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IN   MEMORIAM 
FLORIAN  CAJORl 


Digitized  by  the  Internet  Archive 

in  2008  with  funding  from 

IVIicrosoft  Corporation 


http://www.archive.org/details/collegealgebraOOmetzrich 


COLLEGE   ALGEBEA 


COLLEGE   ALGEBRA 


BY 

WILLIAM   H.    METZLER,   Ph.D. 
EDWARD    DRAKE    ROE,   JR.,  Ph.D. 

Professors  of  Mathematics  in  Syracuse  University 
AND 

WARREN   G.    BULLARD,   Ph.D. 

Associate  Professor  of  Mathematics  in  Syracuse  University 


LONGMANS,    GREEN,    &    CO. 

91  AND  93   fifth   avenue,   NEW  YORK 
LONDON,   BOMBAY,   AND  CALCUTTA 

1908 


Copyright,  1908, 

BY 

LONGMANS,   GREEN,   &  CO. 


r  i  4-v 


INTRODUCTION 

The  facts  of  Algebra  are  of  minor  importance  to  the 
average  individual  and  the  subject  should  not  be  studied 
with  the  acquiring  of  these  facts  as  the  principal  object 
to  be  attained.  Algebra  studied  for  the  mere  body  of  facts 
which  it  contains  is  a  waste  of  time.  These  facts  the  student 
will  of  course  acquire,  but  the  authors  believe  they  should 
come  as  incidentals  to  the  acquiring  of  the  methods  and 
principles  of  the  subject.  The  principal  object,  therefore, 
for  both  teacher  and  student  to  keep  in  mind  is  the  acqui- 
sition, not  of  the  facts,  but  of  the  underlying  methods  and 
principles,  and  we  believe  that  when  this  is  done  the  facts 
will  be  more  intelligently  comprehended  and  better  retained. 

We  have  endeavored  to  develop  the  topics  treated  in 
as  logical  and  scientific  a  manner  as  was  consistent  with 
good  pedagogy.  The  ground  required  for  entrance  to  the 
scientific  courses  of  the  leading  Colleges  and  Schools,  or 
that  required  in  the  freshman  year  by  the  students  in  the 
course  in  arts  has  been  covered,  and  in  addition  the  needs  of 
more  advanced  students  have  been  kept  in  mind.  Rather 
more  ground  is  covered  than  is  laid  down  in  the  require- 
ments for  the  examinations  in  Advanced  Algebra  by  the 
College  Entrance  Examination  Board.  In  any  case  no  diffi- 
culty will  be  experienced  in  omitting  the  extra  parts  if  the 
teacher  so  desires. 

The  major  portion  of  the  book  has  been  used  in  pamphlet 
form  for  several  years  with  good  results  by  the  freshmen  at 
Syracuse  University. 


TABLE  OF  CONTENTS 


CHAPTER   I 
Graphic  Representation  of  a  Function 


Representation  of  a  point 

Distance  between  two  points 

Locus  of  a  moving  point 

Equations  of  the  first  degree 

Equations  of  higher  degree 

Solution  of  equations  by  graphic  methods 

Simultaneous  equations  of  the  first  degree 

Simultaneous  equations  of  higher  degree 

Examples 


1 
2 
3 
4 

6 
10 

10,  11 

11,  12 
2,  6,  8,  9,  10,  11,  12 


CHAPTER   II 

Inequalities 

Definitions  and  notations 13 

Theorems 14-17 

Examples 17-19 


CHAPTER   III 

Ratio,  Proportion,  and  Variation 

Ratio 20 

Definitions 20,  21 

Theorems 21 

Proportion 22 

Definitions 22 

Theorems 22-27 

Variation 28 

Definitions 28 

Graphic  illustrations 28,  29 

Theorems 29,  30 

Examples 21,  26,  27,  30,  31 

vii 


Vlll 


TABLE   OF   CONTENTS 


CHAPTER  IV 

Theory  of  Quadratics 

The  sum  and  product  of  the  roots  . 
Formation  of  equations  with  given  roots 
Factoring  quadratic  expressions 
Solution  of  quadratics  by  inspection 
Nature  of  the  roots  of  a  quadratic  . 

Discriminant 

Geometric  representation  of  roots   . 

Every  quadratic  has  two  and  only  two  roots 

Examples 


PAGE 

32,  33 
33 

34 

35 

36,  37 

38 

38,  39 

41,  42 

35,  42,  43 


CHAPTER  V 
Factor,  Identity,  and  Remainder  Theorems 

Factor  theorem ,44 

Number  of  roots  of  an  equation 45 

Identity  theorem 40 

Identity  of  two  polynomials 4G,  47 

Remainder  theorem 48 

Examples 49 

CHAPTER  VI 
Commensurable,  Incommensurable,  and  Imaginary  Numbers 

Definitions 50,  51,  52 

Theory  of  indices 52-56 

Radicals,  definitions 57,  58 

Reduction  of  surds 58,  59 

Reduction  of  a  mixed  to  an  entire  surd  ......         60,  61 

Addition  and  subtraction  of  surds 62 

Multiplication  of  surds 63-65 

Division  of  surds QQ 

Involution  and  evolution  of  surds 67 

Rationalization        ...........       68 

Properties  of  quadratic  surds 72 

Theorems         ............       72 

Square  root  of  a  binomial  surd 73 

Radical  equations 74-77 


TABLE   OF   CONTENTS 


IX 


Complex  numbers  ..... 

Definitions 

Properties  of  conjugate  complex  numbers 
Identity  theorems  for  complex  numbers  . 
Graphic  representation  of  complex  numbers 


.  79 
.  80 
.  81 
.  81 
82-91 


Examples       56,  57,  59,  60,  61,  62,  63,  64,  05,  67,  69-72,  74,  77,  78,  82,  91-94 


CHAPTER   VII 


Progressions 


Arithmetical  progression 
Definitions 
Sum  of  n  terms 
Geometrical  progression 
Definitions 
Sum  of  n  terms 
Harmonical  progression 
Examples 


.   95 

95-97 
.   96 
.   99 
99,  101 
.  100 
103,  104 
96,  97-99,  100-103,  105-107 


CHAPTER   VIII 
Permutations  and  Combinations 

Definitions,  notations,  fundamental  principle  ....     108,  109 

Permutations 109,  110 

Combinations  ............     113 

Theorems 113-121 

Examples 111-113,  121-123 


CHAPTER   IX 


Binomial  Theorem 


Proof  by  induction  .... 

General  term 

Some  properties  of  binomial  coefficients 
Theorems         ..... 
Expansion  of  a  multinomial     . 
Extraction  of  roots  .... 
Examples 


124,  125 

.  129 
.  130 
132,  133 
134-136 
.  136 
128,  130,  131,  137,  138 


TABLE   OF   CONTENTS 


CHAPTER  X 
Constants,  Variables,  and  Limits 

PAGE 

Definitions 139 

Theorems  ............     140 

Illustrative  examples 140-145 


Value  of 


Examples 


-y" 


u  —  V 


145,  146 
.     146 


CHAPTER   XI 

Series 

Definitions 147 

Theorems 148,  149 

Methods  for  testing  the  convergency  or  divergency  of  a  series        .      151-160 
Examples 160,  161,  162,  163 

CHAPTER   XII 

Undetermined  Coefficients 

General  theory . 

Theorems 

Development  of  an  algebraic  fraction  into  a  series 
Binomial  theorem  for  any  real  exponent 
Decomposition  of  fractions  into  partial  fractions    . 
General  term  in  the  development  of  an  algebraic  fraction 
Summation  of  integral  series 


Examples 


Definitions 

Theorems 

Examples 


.  164 
164,  165 
166-168 
169-172 
173-181 

.  182 
183,  184 
169,  173,  181,  182,  183,  184 


CHAPTER   XIII 
Continued  Fractions 


.      185-195 

.      185-195 

188,  190,  191,  192,  193,  195 


CHAPTER   XIV 


Integral  Solutions  of  Indeterminate  Equations  of  the 
First  Degree 

Particular  and  general  solutions 196,  197 

Examples 197,  198 


TABLE   OF   CONTENTS  xi 

CHAPTER  XV 
Summation  of  Series 

PAGE 

Series  whose  nth  term  is  in  the  form 

(a+w6)[a+(n  +  l)6]---[a+(w+m-l)6] 199 

Series  whose  wth  term  is  of  the  form 

L 200 

Recurring  series 203 

Definition  and  sum  of  7i  terms 203-205 

Generating  function 205 

Finite  differences 206-209 

Interpolation 212-215 

Examples 201,  202,  206,  209-212,  215,  216 

CHAPTER   XVI 

Logarithms 

Definitions 217 

Theorems 217-219 

Change  from  one  base  to  another 220,  221 

Determination  of  logarithms  of  numbers  and  the  use  of  tables        .      221-223 

Tables 224,  225 

Determination  of  a  number  from  its  logarithm 226 

Cologarithms 227 

Computation  by  logarithms,  illustrative  examples  ....      228-231 

Exponential  function 231-241 

Exponential  theorem       ..........     241 

Logarithmic  series 241 

Calculation  of  logarithms 242,  243 

Examples 220,  230,  231,  243 

CHAPTER  XVII 

Determinants 

Definitions  and  notations 244-247 

Theorems 248-252 

Minors 253,  254 

Expansions 250,  254,  255 

Solution  of  linear  equations 256-259 


Xll 


TABLE   OF   CONTENTS 


Product  of  two  determinants 
Examples 


PAGE 

.     259,  260 
247,  248,  263,  261 


CHAPTER  XVIII 
Theory  of  Equations 

Significant  term  of  a  polynomial  in  the  case  of  large  and  small  values     262,  263 

/  Development  of  a  function 263,  264 

Continuity 265 

Theorems 266 

Descartes'  rule  of  signs 267 

Complex  roots  enter  in  pairs 268 

Number  of  real  roots  between  a  and  h 268-270 

Relations  between  coefficients  and  roots 270 

Cube  roots  of  unity 271 

.  Symmetric  functions        .........     271,  272 

!„  Factoring  of  symmetric  and  related  expressions      ....      272-274 

Transformation  of  equations 275-277 

Contracted  division 277 

Geometric  interpretation  of /'(ic) 280,  281 

Rolle's  theorem •     282 

Multiple  roots  of /(a:) 283,284 

The  signs  of /(x)  and /(a:)  on  passing  through  a  root  oif(x)  =  0  .      284,  285 

Transformed  equation  having  one  term  less 285,  286 

Solution  of  the  cubic 287-289 

Discriminant  of  the  cubic 289 

Solution  of  the  biquadratic 290-292 

Sturm's  theorem 292-295 

Solution  of  numerical  equations 296,  297 

Horner's  method 297-300 

Examples         .        .        .264,266,267,271,272,278,279,280,282,300,301 


CHAPTER   XIX 
Miscellaneous  Topics 


Mathematical  induction  .         .         .        .         .        . 

Limits 

Theorems  concerning  infinitesimals  and  infinites    . 
Theorems  on  limits  ...... 

Convergency  and  divergency  of  some  particular  series 


302-305 

.  305 

306-309 

309-311 

.  312 


TABLE   OF   CONTENTS  xiii 

PAGE 

Limits  of  ratios 313 

Theorems  on  convergence 315 

Examples 303,  305,  317 

Product  of  two  infinite  series 320 

Vandermonde's  theorem 321 

Binomial  theorem  for  any  index 321-323 

Complex  variable  as  a  function  of  its  modulus  and  argument  .         .     323 

De  Moivre's  theorem 324 

Continuity  of  f{z) 325 

Geometric  representation  of /(«) 325 

Isogonality  of /(s) 326-329 

Failure  of  isogonality 329 

Fundamental  proposition  of  algebra 330,  331 

Index      .        .        « 333 


COLLEGE  ALGEBRA 


CHAPTER   I 


GRAPHIC  REPRESENTATION  OF  A  FUNCTION 


o 


X' 


1.  Representation  of  a  Point.  If  a  point  is  to  be  located, 
it  must  be  clone  with  reference  to  some  known  or  fixed 
positions,  usually  with  reference  to  known  lines.  Thus  to 
determine  the  position  of  a  point  in  a  plane,  let  the  two 
lines  X'X  and   Y^Y  intersect 

XT" 

at  right  angles  in  the  point  0. 

Then  if  we  are  given  that  a 
point  P  is  h  units  distant  from 
X' X  and  a  units  distant  from 
Y'  F,  it  is  located  as  one  of 
four  points.  The  fact  that  it 
is  h  units  distant  from  X'X 
limits  it  to  two  lines,  one  on 
either  side  and  parallel  to  X'X^ 
and  at  a  distance  of  b  units  from 
it.  Similarly,  the  fact  that  it 
is  a  units  distant  from  Y'  Y 
limits  it  to  two  lines  parallel  to  Y'  Z",  one  on  either  side  and 
a  units  distant  from  it.  If  these  two  limitations  are  imposed 
at  once,  the  point  P  is  one  of  the  four  intersections  of  these 
four  lines. 

Definitions.  If  now  we  suppose  distances  measured  in  the 
directions  OX  and  OY  affected  with  the  positive  sign  and 

B  1 


Fig.  1. 


2  COLLEGE   ALGEBKA 

distances  measured  in  the  opposite  directions  affected  with 
the  negative  sign,  then  tlie  point  P  can  be  definitely  located. 

The  distances  thus  measured  from  OX  and  OY  and 
affected  with  the  proper  signs  are  called  the  ordinate  and 
abscissa,  respectively,  or  together  tlie  coordinates  of  P. 

The  line  X' X  is  called  the  axis  of  abscissas  or  tlie  x-axis  ; 
and  the  line  Y'  Y  is  called  the  axis  of  ordinates  or  the  y-axis. 
The  point  0  is  called  the  origin.  If  P  is  a  units  to  the 
right  of  the  ?/-axis  and  h  units  above  the  a;-axis  its  coordi- 
nates are  a  and  h  and  the  point  is  denoted  by  (a,  5).  If  it 
is  a  units  to  the  left  of  the  ^-axis  and  h  units  above  the 
ir-axis,  its  coordinates  are  —  a  and  h  and  the  point  is  denoted 
by  (—a,  5).  The  abscissa  and  ordinate  of  a  point  are  usu- 
ally denoted  by  x  and  y  respectively.  Thus  the  point  x  =  a^ 
y  =  h  denotes  the  point  (a,  5),  or  whose  abscissa  is  a.,  and 
whose  ordinate  is  h ;  and  the  point  x=  a  and  y  =  —  h  denotes 
the  point  (a,  —  ^).  A  point  whose  coordinates  are  unknown 
is  usually  denoted  by  (x^  ?/). 

The  parts  of  the  plane  between  OX  and  OY^  OY  and 
0X\  OX'  and  OY' ,  OY'  and  OX  are  called  the  first, 
second,  third,  and  fourth  quadrants  respectively.  The  point 
P  is  in  the  first,  second,  third,  or  fourth  quadrant,  according 
as  (a-th)^  C^cii  ^),  (  — «,  — ^)?  oi'  (^^  — ^)  a-re  its  coordinates. 

EXAMPLES 

1.  Locate  the  points  (2,  3),  (1,  -  2),  (-  3,  0),  (0,  -  1), 
(-3, -4),  (0,0),  (-7,8). 

2.  To  what  does  x=  S  limit  a  point  ?  To  what  does 
x  =  —  4:  limit  a  point  ?  To  what  does  y  =  5  limit  a  point  ? 
To  what  does  y  =  —  7  limit  a  point  ? 

2.  To  express  the  distance  between  two  points  in  terms  of 
the  coordinates  of  those  points. 


GRAPHIC   REPKESENTATIOX  OF   A  FUNCTION 


Let  (a-j,  ?/j),  (2^2,  ^2)  ^®  ^^^®  coordinates  of  the  two  points 

Pv  ^2-  Y  E, 

Then 


or 


V(:C2-^'l)^  +  (j/2-^l)^-* 

If  P2  is  ^^®  origin,   this  becomes 


>x 


Fig.  2. 


3.  To  find  the  equation  of  the  locus  or  path  of  a  point 
moving  according  to  some  law  is  to  find  the  equation 
satisfied  by  the  coordinates  of  every  point  on  the  locus. 
Thus  to  find  the  equation  of  the  locus  of  a  point  (2:,  ?/) 
which  is  always  equally  distant  from  the  two  points  (.r-^,  y^), 
(^2'  ^2)'  ^^^  have 


■VCx  -  x^y  +  (?/  -  ^i)2  =  V(a;  -  x^y  +iy-  y^yf, 

or    2  {x^  -x^x-\-i  {ij^^^  -yi)y  +  ^1^  -  x.^  +  yi^  -yi^^- 

Where  does  this  locus  cut  the  line  joining  the  two  points  ? 

Again,  to  find  the  equation  of  the  locus  of  a  point  which 
moves  so  that  it  always  remains  at  a  distance  of  seven  units 
from  the  origin,  we  have 


or  a;2  +  ?/2=49. 

Of  what  locus  is  this  the  equation  ? 


*  Since  the  distance  between  two  points  is  positive,  we  neglect  the  two- 
fold sign  before  the  radical. 


COLLEGE  ALGEBRA 


In  general,  to  find  the  equation  of  the  locus  of  a  point 
which  moves  so  that  it  always  remains  at  a  distance  of  r 
units  from  a  fixed  point  (a,  /3),  we  have 


or  {x  —  of  +  (y  —  yS)^  =  T^' 

This  is  evidently  the  equation  of  the  circle  whose  radius 
is  r  and  whose  center  is  at  (a,  /3). 

4.  To  'plot  a  point  is  to  locate  the  point  in  the  plane  by 
means  of  its  coordinates,  and  to  plot  a  curve  is  to  trace  it  by 
means  of  its  points  or  otherwise. 

5.  Graphs  of  Equations  of  the  First  Degree.  We  have 
seen  that  the  equation  x=%  represents  a  line  parallel  to  the 
^-axis  and  at  a  distance  of  three  units  to  the  right  of  the 
origin.  It  will  be  shown  by  means  of  graphical  represen- 
tation that  abstract  algebraical  relations  (equations)  between 
two  unknown  quantities  can  be  represented  in  concrete  form 
as  geometrical  curves. 

6.  Let  us  consider  the  equation 

y  =lx  -\-  h. 

Every  pair  of  values  of  x  and  y  which  satisfy  the  equa- 
tion may  be  taken  as  the  coordinates  of  a  point  in  a  plane, 
and  the  assemblage  of  all  points  whose  coordinates  are  solu- 
tions of  this  equation  is  called  the  locus  or  graph  of  the 
equation. 

Let  {x^,  ^j),  (x^,  ^/g),  (rrg,  y^  be  the  coordinates  of  any 
three  points  Pj,  P^,  Pg,  on  this  locus.     Then  w^e  have 

y^  =  Ix^  +  5,  (1) 


GRAPHIC   REPRESENTATION  OF   A   FUNCTION  5 

Subtracting  (1)  from  (2)  and  also  from  (3),  we  have 


^2  -  ^1  =  K^l  -  ^l)' 

^3-^1  =  ^(^3-^1)- 

Dividing  (4)  by  (0),  we  have 

y^  —  y\  —  -^2     ^1 


Vz  -  Vi     H 


—  X. 


O) 
(5) 

(6) 


>x 


Fig.  3. 


J^\-ti  —  X^        Xy, 

HP 2.  =  ^2  -  ^1' 

^^A  =  ^3  -  ^r 

The  triangles  Py^^H  and  P^P^T^  are  right  triangles,  and 
by  equation  (6)  have  their  sides  including  the  right  angles 
proportional,  and  are  therefore  similar,  so  that  the  angle 
KP^P^  =  the  angle  HP^P^,  and  therefore  the  points  P^,  P^^ 
Pg  lie  in  the  same  straight  line.  Since  Pj,  P^,  Pg  are  any 
three  points,  it  follows  that  the  locus  is  a  straight  line. 
From  this  it  is  seen  that  every  equation  of  the  first  degree 


6  COLLEGE   ALGEBRA 

of  this  form  represents  a  straight  line ;  and  the  most  gen- 
eral equation   of  the  first  degree  Ax-\-B'i/-{-  C=0  can  be 

A         0 
written  in  this  form  (^B  ^  0),  thus  y  =  —  —x If  JB  =  0, 

B        B 

the  equation  Ax  +  (7=  0  is  a  line  parallel  to  the  ?/-axis.  It  fol- 
lows that  every  equation  of  the  first  degree  represents  a  straight 
line.  For  this  reason  equations  of  the  first  degree  are  often 
called  linear  equations. 

From  the  fact  that  every  equation  of  the  first  degree 
represents  a  straight  line,  and  since  two  points  determine  a 
straight  line,  in  plotting  an  equation  of  the  first  degree  it  is 
only  necessary  to  plot  two  of  its  points. 

Thus,  for  the  equation 

2  a;  +  3  ?/  =  6 
when  x  =  0,     y  =  % 

when  y  =^^     2;  =  3, 

which  gives  us  two  points  where  the  line  meets  the  coordi- 
nate axes. 

7.  Plot  the  lines  : 

1.  ?/  =  2a;  +  3. 

2.  y  =  x. 

3.  2a;-7?/  +  4  =  0. 

4.  x-[-  y  =  0. 

8.  Graphs  of  Equations  of   Degree  higher  than  the  First. 

Consider  the  equation  x^  -\- y"^  = -^^ .  We  know  from  3  that 
this  is  a  circle  with  center  at  the  origin  and  with  radius  7. 
From  the  equation  itself  we  can  easily  see  that  the  locus  is 
symmetrical  with  respect  to  the  axis  of  x ;  for  if  we  give  x 
any  value,  there  will  be  two  values  of  y  equal  but  opposite 
in  sign.  Similarly,  the  locus  is  seen  to  be  symmetrical  with 
respect  to  the  ^-axis. 


GRAPHIC  REPRESENTATION  OF  A  FUNCTION 


Again,  consider  the  equation  ?/2  =  4  a;,  and  we  see  that  the 
locus  is  symmetrical  with  respect  to  the  2;-axis,  but  is  not 
with  respect  to  the  ?/-axis. 

9.  If  we  have  two  equations  in  x  and  y^  each  will  repre- 
sent a  curve,  and  the  common  solutions  of  the  two  equations, 
considered  as  simultaneous,  will  represent  the  coordinates 
of  the  intersections  of  the  two  curves. 

Thus  the  circle  x^  +  y^ 
=  25  and  the  line  x-{-  y  ='Z 
intersect  in  the  two  points 
whose  y  coordinates  are 
given  by  the  equation 


Y 


or 


(3  -  ^)2  ^yl  =  25, 

3±V4T 


y^ 


The   X   coordinates  of  the 

.  ,  StVTi 

same    points  are , 


\ 

' 

^ 

X 

\ 

/ 

\ 

\ 

/ 

\ 

\ 

' 

\ 

\ 

0 

\ 

\ 

\ 

V 

\ 

/ 

\ 

\ 

/ 

\ 

^_ 

-^ 

X 


Y' 
Fig.  4. 


found  by  substituting  the 

values  of  y  in  the  equation  of  the  line. 

10.    Expanding  the  equation  of  the  circle  obtained  in  3, 

we  have 

x^  +  ?/2_  ^2ax-1^y-^  iC-  +  jS^- -  r2  =  0, 

which  shows  that  in  order  that  the  equation  of  the  second 

degree 

ax^  -\-  2  hxy  +  hy^  -\-  2  yx  +  2fy  -\-  c  =  0 

may  represent  a  circle,  we  must  have  a  =  h,  and  7i  =  0;   then 
the  equation  becomes,  after  dividing  by  a, 

9        9      2  ^        2/        e      ^         X  / 
a  a  a 


8  COLLEGE   ALGEBRA 

Comparing  these  two  equations,  we  have 

a  a  a 

which  shows  that  the  coordinates  of  the  center  are  —  -  and 

f  .    . 

— -,  and  that  the  radius  is 
a 

r  —  -  V/^  +  ^^  —  ^^. 
a 

Thus,  to  find  the  center  and  radius  of  the  circle 


we  have         «  =  3,  yS  =  4,  and  r  =  V9  +  16  —  16  =  3. 

11.  EXAMPLES 

1.  Plot  the  points  (-  5,  7),  (-  1,  -  10),  (3,  -  5). 

Find  the  distance  between  the  points : 

2.  (2,  3)  and  (5,  7). 

3.  (4,  6)  and  (2,  3). 

4.  Obtain  the  formula  for  the  distance  between  the  points 
Pj  (^r  y\)  ^^^  -^2  (^2'  y^  when  they  are  situated  anywhere 
and  not  both  in  the  first  quadrant  as  in  2. 

Thus,  for  Pj  in  the  second  quadrant  and  P^  in  the  fourth 
quadrant  we  should  have  as  before. 


■2         iK/rrt  2 


P^P^  =  MP^-\-MF^, 
MP^  =  MQ+QP^ 
=  -QM-{-QP^, 


GRAPHIC   REPRESENTATION  OF   A   FUNCTION 


9 


Fig.  5. 

since  QM  taken  in  the  opposite  direction  must  be  regarded 
as  the  negative  of  MQ, 

or        MF^  =  —  x-^-\-  x^  =  x^  —  x^,     (  QM=  x^,  QP^  =  x<^.) 

Similarly,  P^M=  P^R  +  RM 

=  -  RP^  +  RM 

=  -^1  +  ^2  =  ^2-^1- 


Therefore 


I\P^  =  ix^-x,y  +  iy^-y{)\ 


P^P^  =  V(x,  -  x{)'^  +  iy^  -  y,)\ 


Find  the  distance  between  the  points : 

5.  (2,  -4)  and  (4,  6).  7.    (2,  3)  and  (-7,  -10). 

6.  (2,  -  3)  and  (3,  -  2).  8.    (-  3,  6)  and  (2,  -  4). 

Find  the  equation  of  the  locus  of  a  point  which  is  equally 
distant  from  : 

9.    (2,  3)  and  (5,  7).  10.    (2,  -5)  and  (-3,  4). 

11.    (-3,  6)  and  (2,  -4). 


10  COLLEGE   ALGEBRA 

Find  the  locus  of  a  point  which  is  always  at  the  distance  of: 

12.  5  units  from  the  origin. 

13.  3  units  from  the  origin. 

14.  9  units  from  the  origin. 

15.  Plot  the  locus  in  each  of  the  problems  9-11. 

16.  What  angle  does  the  locus  make  with  tlie  lines  joining 
the  two  fixed  points  ? 

17.  Through  what  point  in  the  line  joining  the  two  fixed 
points  does  the  locus  pass  in  each  case  ? 

Plot  the  lines : 

18.  3  a:  — y  +  5  =  0. 

19.  2x-\-^+  b  =  0. 

20.  x-\-S  y  =  0.. 

21.  Plot  the  curve  y^=S6  —  x^. 

Find  the  coordinates  of  the  point  of  intersection  of : 

22.  The  line  2:4-2y— 3  =  0  and  the  curve  y^  =  4:  x. 

23.  The  line  12  a:  —  5  ?/  =  169  and  the  circle  oP'-\-  y^  —  169. 

Find  the  center  and  radius  of  each  of  the  circles  represented 
by  the  following  equations : 

24.  a;2  +  ?/2  _  4  a:  +  10  ?/  -  71  =  0. 

25.  a:2  +  ^2_|_2^  +  8^  +  16  =  0. 

26.  2;2^^2_^_l_l()^_^25  =  0. 

SOLUTIONS   OF  EQUATIONS   BY   GRAPHICAL  METHODS 

12.  Simultaneous  Equations  of  the  First  Degree.  We  have 
seen  that  to  every  equation  of  the  first  degree  in  x  and  y 
there  corresponds  a  straight  line ;  and  that  the  infinite  num- 
ber of  pairs  of  values  of  x  and  y  which  satisfy  this  equation 


GRAPHIC  REPRESENTATION  OF  A  FUNCTION 


11 


are  the  coordinates  of  the  points  on  the  line.  If  we  have 
two  equations  of  the  first  degree  in  x  and  y,  it  is  apparent 
that  the  common  solution  of  the  two  equations  is  represented 
by  the  point  of  intersection  of  the  two  lines. 

A  graphical  solution  of  simultaneous  equations  of  the  first 
degree  in  x  and  y  consists  in  plotting  the  lines  and  determin- 
ing the  point  of  intersection  by  measurement.  It  is  evident 
that  the  solution  thus  found 
is  only  approximate  and  that 
the  degree  of  approximation 
obtained  depends  upon  the 
accuracy  of  the  measure- 
ment. This  method  is  often 
employed  by  engineers. 

Thus   to   solve   the   equa- 
tions ,    -,         rs 

^  — y +  1  =  0, 
a;  +  7/  -  5  =  0, 


we  have,  plotting  the  lines 
and  measuring,  2:= 2  and  j/= 3. 


Y 

1 

\. 

/ 

/ 

/ 

\ 

,^ 

/ 

\ 

vV 

\ 

^\ 

/ 

\ 

^/ 

\ 

/ 

\ 

fv 

/r 

\ 

^ 

/ 

1 

\ 

^^ 

/ 

\ 

^n 

/ 

\ 

X 

Y' 


Fig.  6. 


32;  +  4?/-17  =  0. 


XI 


Solve  graphically  the  following  equations : 

1.  2:+ 2?/  — 3  =  0,  3. 
2a;-?/-l  =  0. 

2.  4  :?;  +  7  ?/  +  25  =  0,  4.    7  re  -  3  ?/  +  3  =  0, 
32:-2?/  +  ll  =  0.  4a;-o?/=0. 

13.    To  solve  graphically  two  equations  of  the  form 
rj^j^yi^  2gx  +  2fy  +  c  =  0, 
and  Ax  +  By  +  C=  0, 

we  find  the  center  and  radius  of  the  circle  represented  by 
the  first  equation  and  draw  the   circle,  then  plot  the  line 


12 


COLLEGE   ALGEBRA 


represented  by  the  second  equation,  and  measure  the  coordi- 
nates of  the  points  of  intersection. 

Thus,  if  we  have  the  equa- 
tions 

and  a;  —  ?/  +  7  =  0, 

we  have 

«  =  -4,  /3  =  3, 


X 


' 

k 

/ 

r'<^ 

/^ 

* 

,y 

r 

/ 

"^ 

/C 

/ 

/ 

V 

/ 

X 

^ 

/ 

Fig.  7. 


O 


and  r  =  Vl6  +  9-21  =  2, 

and  the  figure  (7)  for  the 
solution. 
X       Solve  graphically  the  fol- 
lowing equations : 

1.  a;2  +  ?/2-|.10:z:-8i/-8  =  0,  a;=j/. 

2.  a;2+i/2_^12a;-f-14?/-15=0,  a:+^-f5  =  0. 

14.    To  solve  graphically  two  equations  of  the  form 
a:2  +  ?/2  _f.  2^:^:  +  2/j/  +  c  =  0, 
and  2:2  +  /  4-  2^3;  4.  ^fy  +  c'  =  0, 

we  describe  the  circles  represented  by  the  two  equations  and 
measure  the  coordinates  of  the  points  of  intersection. 
Solve  graphically : 

1.    a:2  +  y2_4^_6^_3  =  0, 
2     2:2  +  ^2  ^  25, 

3.    a:2+z/2- 6?/  =  7, 
a:2  +  ?/2  —  4  ?/  =  7. 


CHAPTER   II 

INEQUALITIES 

15.  Definition.  Any  numher  a  is  said  to  he  greater  than  any 
other  number  h  when  a—  h  is  positive ;  and  any  number  a  is 
said  to  be  less  than  b  wheii  a—  b  is  negative. 

Thus  1  is  greater  than  —5,  because  1  — (— 5),  or  6,  is 
positive ;  and  —  7  is  less  than  —  3,  because  —  7  —  (—  3),  or 
—  4,  is  negative. 

In  accordance  with  this  definition  zero  must  be  regarded 
as  greater  than  any  negative  number. 

'  16.  Definition.  In  algebra  an  inequality  is  defined  to  he  the 
statement  that  two  numbers  are  unequal.  From  this  definition 
it  follows  that  one  number  is  greater  than  the  other,  and 
the  statement  of  inequality  often  includes  the  information 
as  to  which  is  the  greater.  In  the  discussion  which  follows 
we  shall  usually  treat  inequalities  of  the  latter  type. 

17.  Notation.  The  symbols  >,  <,  :9b  are  used  as  signs  of 
inequality.  In  the  case  of  >  and  <  the  opening  is  toward 
the  greater  number.  Thus  a  >  5  is  read  "  a  is  greater  than 
5."  A  stroke  through  the  sign  =,  >,  or  <  negatives  its 
significance.     As  examples  of  the  use  of  these  symbols  we 

^^^^^  5>3,     2<6,     2:^3,     ^>Q,     4<3. 

18.  Definition.  Tlie  members  of  an  inequality  are  the  num- 
bers compared. 

13 


14  COLLEGE   ALGEBRA 

The  inequalities  a>h^  c  >  t?  are  said  to  subsist  in  the 
same  sense,  and  the  inequalities  a > 5,  c<^d  are  said  to  subsist 
in  the  opposite  sense. 

19.  Theorem.     If  a>h,  and  h>c,  then  a'>c. 
Proof.  a  —  h  'd^wd  b  —  c  are  both  positive. 
Therefore    (a  —  h')-\-(h  —  c'),  ov  a  —  c  is  positive  and  a>  c. 

20.  Theorem.  If  the  same  number  be  added  to  or  sub- 
tracted from  each  member  of  an  inequality^  there  results  an 
inequality  subsistijig  in  the  sa7ne  seiise. 

Proof.  Suppose  a>b;  then  by  definition  a  —  ^  is  posi- 
tive ;  therefore  the  numbers  a  -]-  c  —  (b  +  c)  and  a  —  c  — 
(^  —  c)  are  both  positive,  since  each  is  equal  to  a—  b. 

Hence  a  -\-  c>b  -{-  c, 

and  a  —  c>b  —  c. 

Corollary.  Any  term  of  an  inequality  may  be  transposed 
from  one  member  to  the  other  by  changing  its  sign. 

Thus  in  the  inequality  a  +  5  >  c  we  may  subtract  b  from 
each  member  and  obtain  a> c  —  b. 

21.  Theorem.  If  the  members  of  an  inequality  be  iyiter- 
changed,  the  sig7i  of  inequality  must  be  reversed. 

Proof.  If  a>b,  a  —  b  is  positive,  b  —  a  is  negative,  and 
therefore  b<a. 

22.  Theorem.  If  the  signs  of  both  members  of  an  inequality 
be  changed,  the  sign  of  inequality  must  be  reversed. 

Proof.     Suppose  a>b; 

then  -h>  -a,  (20,  Cor.) 

and  therefore  —  a<  —  b.  (21) 


INEQUALITIES  15 

23.  Theorem.  An  inequality  ivill  subsist  in  the  same  sense 
after  each  member  has  been  multiplied  or  divided  by  the  same 
positive  number. 

Pkoof.  Suppose  a>b  ;  then  by  definition  a  —  b  is  posi- 
tive, and  therefore,  if  m  be  positive,  m(^a  —  b)  and  —(^a—b^ 

a       b  ^ 

are  positive,  and  hence  am  >  bm  and  —  >  — . 

m     m 

24.  Theorem.  If  the  reciprocals  of  both  members  of  an 
inequality  between  positive  numbers  be  taken,  the  sign  of  in- 
equality must  be  reversed. 

Proof.  Suppose  a>b.  Then  dividing  both  members  of 
the  inequality  by  ab.,  we  have 

i>-,  (23) 

0     a 

and  therefore  -  <  -•  (21) 

a     b 

25.  Theorem.  If  the  members  of  an  inequality  be  multiplied 
or  divided  by  the  sa7ne  negative  7iumber,  the  sign  of  inequality 
must  be  reversed. 

Proof.     Suppose  a>b;  then  —a<-b.  (22) 

—  ac<  —  be,  and < (23) 

c  c 

26.  Theorem.  If  the  corresponding  members  of  inequalities 
subsisting  in  the  same  sense  be  added,  the  sums  tvill  be  unequal 
in  the  same  sense. 

Proof.     Suppose  a-^  >  b-^,  «2  >  ^v  ^3  >  ^3'  " "'»  ^m  >  ^m- 
Then   by   definition    a^  —  b^,    a^—b^^    a^  —  b^,   •••,  ajj^—bj^, 
are  positive. 

Therefore     ^^  —  ^^  +  a^  —  ^2  +  ^3  ~  ^3  +  ' "  +  ^w  ~  ^»i   ^^   posi- 
tive, and  rtj  +a2  +  a3+  •••  -f-  a,,,>b^-\-b^-\-b^-\ [-b^. 


16  COLLEGE   ALGEBRA 

27.  Theorem.  If  the  corresponding  members  of  inequalities 
between  positive  numbers  and  subsisting  in  the  same  sense  be  multi- 
plied together^  the  products  will  be  unequal  in  the  same  sense. 

Proof.     Suppose  a-^  >  Jj,  a^  >  b^^  a^>b^,  •  •  •,  a,„  >  b,^. 
Then  a^a^  >  a^^^  (23) 

and  «2^i  ^  ^1^2* 

Therefore  a^a^  >  b^b^^  (19) 

which  shows  that  the  theorem  holds  for  any  two  inequalities. 
Therefore  it  holds  for  the  inequalities 

a^a^  >  b-J)^^ 
and  ^3  >  ^3, 

and  hence  a-^a^a^  >  b^b^b^, 

and  by  repeating  this  process  we  arrive  at  the  general  result 

a^a^a^  •  •  •  ^m  >  ^1^2^3  '  * '  ^tn' 

28.  It  follows  from  27  that  if  a  and  b  are  positive  and 
a>b,  then  a" > V, 

and  therefore  a~^  <  5~%  (24) 

where  n  is  any  positive  integer. 

29.  The  subtraction  of  corresponding  members  of  two 
inequalities  subsisting  in  the  same  sense  does  not  necessarily 
give  an  inequality  subsisting  in  the  same  sense.  Likewise  the 
division  of  the  members  of  one  inequality  by  the  corresponding 
members  of  another  inequality  subsisting  in  the  same  sense 
does  not  necessarily  give  an  inequality  subsisting  in  the  same 
sense.  The  truth  of  these  statements  is  readily  seen  by  con- 
sidering the  inequalities  6  >  4 

and  3>1. 

Subtracting  member  from  member  would  give  3  >  3. 

Dividing  member  by  member  would  give  2  >  4. 


INEQUALITIES  17 

30.  Theorem.  If  two  numhers  are  unequal^  the  sum  of  their 
squares  is  greater  than  twice  their  product. 

Proof.  Let  a^b;  then  (a  —  by  >  0,  since  the  square  of 
any  positive  or  negative  number  is  always  positive. 

Therefore  a'^  -  2  ab -{- b'- >  0 

and  «2  +  52  >  2  ab.  (20,  Cor.) 

We  shall  call  this  the  Fimdartieyital  Inequality. 

31.  The  principle  involved  in  the  last  tlieorem  may  be 
extended  to  expressions  of  a  degree  higher  than  the  second. 
Thus  we  may  have  the  theorem  :  If  two  positive  numbers  are 
imequal^  the  sum  of  their  cubes  is  greater  than  their  sum  multi- 
plied by  their  product. 

Proof.     From  the  result  of  30  we  have 

a^^ab-\-b'^>ab.  (20,  Cor.) 

Therefore  {a^  -  ah  +  b'^^^a  +  ^)  or  a^^b^>  ab(a  +  b).     (23) 

By  similar  processes  corresponding  theorems  may  be  obtained 
for  the  higher  degrees. 

32.  EXAMPLES 

1.  Find  the  limit  of  x  in  the  inequality  ^  x  -{- 1  >  --\-  -  - 

Multiplying  both  members  of  the  inequality  by  15,  (23) 
we  have  752:  + 105  >  5  a:  +  12, 

or  70  2:>-93,  (20) 

and  ^^~1'  ^^^) 

2.  Find  the  limits  of  x  when  the  inequalities 
x-^>S-{-~   and   1<     ^ 


4  4  x-d 

subsist  simultaneously. 


18  COLLEGE   ALGEBRA 

From  the  first  inequality  we  find  x>5  ;  hence  x—  S  is  posi- 
tive^ and  therefore,  from  the  second  inequality, 

1>^,  (24) 

S>x-S,  (23) 

6>x,  (20) 

and  therefore  6>x>5. 

3.  Find  the  limit  of  x  in  the  inequality 

(62;  -  5)(2:  +  4)>  (3:?;  +  2)(2  2;  +  T). 

4.  Find  the  limits  of  x,  given  that 

(2  +  3  a;)  (1  -  a;)  +  3  >  2  a;  -  3  a;2, 

and  (3  2^  +  1)  (2;  +  1 )  -  17  2:  >  (3  2;  -  2)  (a;  -  5)  +  10. 

In  the  following   problems  the  letters    are  supposed  to 
represent  positive  and  unequal  quantities. 

5.  Which  is  the  g-reater,  or  — ? 

^  2  a-\-b 

6.  If  «2  _j_  52  _.  ][  ^YLd  x"^  -\-  y^  =  1,  sliow  that  ax-\-hy  <  1. 

7.  If    a? -{- 1)^ -\- c^  =  1    and    x^ -\- y"^ -\- z^  =  1^     show     that 
ax  -[-hy  +  cz<.  1. 

8.  Show  that  ah(^a  +  ^)  +  hc(h  +  c)  +  (?«(6'  +  a)  >  6  ahc. 

9.  Show  that  (a  +  J)  (5  +  (?)((?  +  a)  >  8  a6(?. 

10.  Prove  a^  +  52  +  6'^  >  a6  +  ^(?  +  c^- 

11.  Show  that  (^a  +  h  —  cy+i^c  +  a  —  hy+ih  +  e  —  ay>  ah 
+  be  -\-  ca. 

12.  The  sum  of  any  positive  fraction  and  its  reciprocal  is 
greater  than  2. 

13.  Show  that  a2J2  _^  j2^2  _^  ^2^2  ->  ^j^^^^  _|.  5  _}_  ^). 

14.  Prove  ^3  4- 2^*3^3  «52. 


INEQUALITIES  19 

15.  Prove  a^  +  b^  >  a%  +  ah^. 

16.  Show   that     2  (a^  +  h"^  +  ^3)  -^  ah  (^a  +  h^ -{-  he  (h  +  c) 
-\-  ca(^c  +  a). 

17.  Show  that  rt^  +  ^^3  _|_  ^3  >  3  «5(?. 

18.  Show  that  2  ^3  +  3  53  >  4  a52  +  a^J. 

19.  Show  that  (x^y  +  ^/^^  +  zH)  (xi/  +  ^s;^  +  rf)  >  9  .-rV^s. 


CHAPTER   III 

RATIO,  PROPORTION,  AND  VARIATION 

RATIO 

33.  Definition.  If  a  and  h  are  two  quantities  of  the  same 
kind^  the  ratio  of  a  to  h  is  the  quotient  of  a  divided  hy  h. 

If  the  ratio  of  a  to  6  is  r,  then  a  =  hr^  which  shows  that  a 
is  r  times  5. 

The  quantities  a  and  h  may  be  concrete,  but  their  ratio  is 
abstract. 

34.  The  ratio  of  a  to  ^  will  be  expressed  as  -  or  as  a  :  h. 

0 

The  quantities  a  and  h  are  called  the  ter77is  of  the  ratio.     The 
quantity  a  is  called  the  antecedent  and  h  the  consequent. 

35.  Definitions.     The  ratio  will  be  one  of  greater  inequality^ 

a  imit  ratio,  or  one  of  lesser  inequality  according  as  a  =  b. 

a  o 

If  we  multiply  together  the  ratios  -  and  -,  the  resulting 

6  d 

ft  p  •  • 

ratio  -—  is  said  to  be  the  ratio  compounded  of  the  two  ratios 
hd 

a  c  a  o 

-  and  -,  or  the  compound  ratio  of  -  and  -• 
0  d  0  d 

EXAMPLES 

1.  Find  the  ratio  compounded  of  |  and  |. 

2.  Find  the  ratio  compounded  of  the  compound  ratio  of  ^ 
and  |,  and  1^  and  |. 

20 


RATIO,   PROPORTION,   AND  VARIATION  21 

36.  Definitions.     The   ratio  compounded   of   the   ratio    - 

2  ^ 

with  itself  is  —  and  is  called  the  duplicate  ratio  of  -.     The 

(1  ft 

compound  ratio  —  is  called  the  triiylicate  ratio  of  -  • 

(T  0 

37.  Definitions.     The  ratio  is  called  the  subdupUcate 

fl  CI  •        •  ft 

ratio  of  -,  and  —  is  called  the  ^ubtriplicate  ratio  of  -• 

EXAMPLES 

1.  Find  the  duplicate  ratio  of  |^f ,  the  subdupUcate  ratio  of  J^. 

2.  Find  the  triplicate  and  subtriplicate  ratios  of  ^-^, 

38.  Definition.  The  ratio  of  6  :  a  is  called  the  inverse  of  the 
ratio  a\h. 

39.  Theorem,  i?^  a  series  of  finite  ratios  ivliich  are  not  all 
equals  hettveen  positive  nu7nhers,  the  ratio  of  the  sum  of  the 
antecedents  to  that  of  the  consequents  is  less  than  the  greatest, 
and  greater  than  the  least  of  these  ratios. 

Proof.     Let  the  ratios  ^,  -^,  -^,  •••,—,  be  denoted  by 

Vl        V^        Vg  v„ 

^r  ^2,  T-g,  •••,  r„,  the  least  of  these  by  r  and  the  greatest  by  B. 
Then  ^  =  r^, 


Uc 

'_    /yi 

2' 


^-2 

—  =  r, 


V 


whence,  by  clearing  each  of  these  of  fractions  and  adding  the 
corresponding  members  of  the  resulting  equations  we  have 
the  equality 

Wi  + 1^2  +  ^3  +  •  •  •  +  ^«  =  ^1^1  +  ^'2^'2  +  ^'3^3  +  •  •  •  +  r„v,,. 


22  COLLEGE   ALGEBRA 

If  in  place  of  the  right  member  we  form  expressions  con- 
taining R  and  r  instead  of  r^,  r^,  •••,  r,,,  it  is  obvious  that  the 
resulting  expressions  are  greater  and  less,  respectively,  than 
the  left  member  of  the  equation,  that  is, 

whence  dividing  by 

we  obtain         R  >    ^        ^  — ^-^ — -  >  r, 

which  proves  the  theorem. 

PROPORTION 

40.  Definition.  Four  quantities  a,  J,  c,  d^  such  that  the  ratio 
a  :  h  is  equal  to  the  ratio  c  :  d^  are  said  to  he  in  proportion  or 
to  form  a  proportion.  It  is  to  be  observed  that  the  quantities 
a  and  b  must  be  of  the  same  kind  and  that  c  and  d  must  be 
of  the  same  kind,  but  are  not  necessarily  of  the  same  kind  as 
a  and  h. 

41.  Definitions.  In  the  proportion  a  :  h  =  e  :  d^  a  and  d  are 
called  the  extremes,  and  b  and  c  are  called  the  means, 

THEOREMS   IN   PROPORTION 

42.  In  any  proportion  the  product  of  the  means  is  equal  to 
the  product  of  the  extremes. 

Let  the  proportion  hQ  a  i  b  =  c  :  d, 

a      c 

Clearing  of  fractions,  we  have  ad=bc. 

43.  In  any  proportion  the  terms  are  in  proportion  by  alter- 
nation, that  is,  the  means  or  the  extremes  can  be  interchanged. 


RATIO,   PROPORTION,   AND  VARIATION  23 

Given  the  proportion  a  :  h  =  c  :  d, 

a      c 

Multiplying  both  members  by  -,  we  have  ~  ~  j* 

As^ain,  multiply  both  members  by  -  and  we  have  -  =  -  • 

a  ha 

44.  In  any  proportion  the  terms  are  in  proportion  hy  in- 
version, that  is^  the  terms  of  each  ratio  can  he  interchanged. 

a      c 
Given  the  proportion,      i  —  ~f 

(J  Cv 

whence,  -  =  — , 

a      c 

h      d 

h      d 
or,  -  =  — 

a      c 

45.  -Zf  four  quantities  are  in  proportion^  they  are  in  propor- 
tion hy  addition,  that  is^  the  sum  of  the  first  tivo  is  to  the  second 
as  the  swn  of  the  second  tivo  is  to  the  fourth. 

a      e 


Given  the  proportion,  7  =  -, 

0      d 


whence, 

h            d 

or, 

a-\-h      c  A-  d 
h            d 

Again, 

h     _    d 

a-^h      e+  d' 

nr 

a             c 

Ui, 

a-}-h      c  -\-  d 

Ao-ain. 

a-\-h      c  -{-  d 

(44) 


multiplying  by  |  =  | 


(44) 


a  c 

Let  the  student  state  the  theorem  for  each  of  these  forms. 


24  COLLEGE   ALGEBRA 

46.  If  four  quantities  are  in  proportion^  they  are  in  propor- 
tion hy  subtraction,  that  is^  the  difference  of  the  first  two  is  to 
the  second  as  the  difference  of  the  second  two  is  to  the  fourth. 

a      G 
Given  the  proportion,       t  =  -^^ 

0      d 

whence,  -  —  1  =  -  —  1, 

0  d 

a—  h      c  —  d 

(If* 

Multiplying  both  members  of  this  by  -  =  -,  we  get 

0      d 

a  c 


a  —  h      c  —  d 

Finally,  ^^  =  ^^.  (44) 

a  c 

Many  authors  use  the  terms  composition  and  division  for 
what  we  have  called  addition  and  subtraction. 

47.    If  four  quantities  are  in  proportion^  they  are  in  propor- 
tion hy  addition  and  subtraction. 

Given  the  proportion,  a  :  h  =  c  :  d, 

then,  a_±h^^±d      ^^^  ^^^-^ 

h  d 


a—  h      c  —  d 


Dividing  (1)  by  (2),  we  have 


b  d 

a  -\-b      c  -\-  d 


(2)  (46) 


a  —  b      c  —  d 


48.    The  quantities  a,  ^,  c,  d,  •••,  are  said  to  be  in  continued 
proportion  ii  a:b  =  b  :  c  =  c  :  d  =  "•. 


RATIO,   PROPORTION,   AND   VARIATION  25 

49.  If  a  :h  =  h  :  c,  then  h  is  called  a  mean  proportional  to 
a  and  <?,  and  c  is  called  a  third  proportional  to  a  and  h. 

50.  If  f  =  ^  =  ^=f,  then  ^"^^"["^"j"f  is  equal  to   any 

b      d     J      h  b-\-  d  -f-/  +  /i 

one  of  the  ratios. 


For  let                   «  _  ^  _  ^  _  ^  _  ^  . 
b      d     f     h        ' 

then                                                a  =  br, 

(1) 

c  =  dr, 

(2) 

e=fr, 

(3) 

g  =  Ar. 

(4) 

Adding  (1),  (2),  (3),  (4),  we  have 

a^-c  +  e  +  g  =  (b  +  d+f+  h)r, 

^                      a-\-  c  -\-e  +  a             a      e 
whence,             ,    '           , — f  =  r  =  - =  -  =  •••. 
b  +  d^f-\-Ji             b      d 

K1      Tf                             a      c      e      g 

,  3/3  a^c  +  2  ace  +  5  g^  +  -I  aeg 

then  \/  ^  .^2^  +  2  ^»cf/  +  5/3  +  4  bfh 

is  equal  to  any  one  of  these  ratios. 

For  let  «  _  £  _  ^  _  .£  =    . 

b~'d~f~h~^' 

then                                          a  =  5r,  (1) 

c  =  dr,  (2) 

g  =/r,  (3) 

g  =  hr.  (4) 

Squaring  (1)  and  multiplying  the  result  by  three  times 

(2)  we  have                        Sa^c=Sb''drK  (5) 


26  COLLEGE   ALGEBRA 

Multiplying  (1),  (2),  (3)  together  and  multiplying  the  result 
by  2,  we  have  2  6«e^  =  2  hdfA  (6) 

Cubing  (3)  and  multiplying  the  result  by  5,  we  have 

^e^=^fh^^,  (7) 

Similarly,  4  aeg  =  4  bfhr^,  (8) 

Adding  (5),  (6),  (7),  (8),  we  have 

3  A  +  2  ac^^^  +  5  ^3  +  4  a^^  =  (3  hH  +  2  J^f  +  5/^  +  4  5/A)r3. 

,^,,                     3  A  +  2  a^e  +  5  e^  +  4  a6?(7        o 
Whence  Tmr, t^-t^tt. — i^t^ rr:^  =  ^  ? 


3ra  +  266?/+5/3  +  4/>/A 


3  a%  +  2  ac£?  +  5  e^  +  4  (^ei^  _     _ci  _  c  _ 
^^'  ^  3  526?  +  2  5(^/  +  5/3  +  4  ^//i  ""  ^  ~  5  ~  ^  ~  "  * ' 

From  the  mode  of  obtaining  this  result  it  is  apparent  that 
a  much  more  general  result  might  be  obtained  on  observing 
that  the  numerator  of  the  fraction  is  homogeneous  in  the 
antecedents  and  the  denominator  is  homogeneous  in  the  con- 
sequents and  of  the  same  degree  as  the  numerator  and  that 
the  degree  of  either  is  the  same  as  the  index  of  the  root.  It 
is  also  to  be  observed  that  coefficients  of  corresponding  terms 
in  the  numerator  and  denominator  may  be  any  numbers  as 
long  as  they  are  the  same. 

52.  EXAMPLES 

1.  What  is  the  duplicate  ratio  of  3  :  4  ?  the  subduplicate 
ratio  of  36  :  25  ?  the  subtriplicate  ratio  of  1728  :  27  ? 

2.  Two  numbers  are  in  the  ratio  of  2  to  3,  and  if  9  be  added 
to  each  they  are  in  the  ratio  of  3  to  4.      Find  the  numbers. 

3.  Show  that  the  ratio  a:h  \^  the  duplicate  of  the  ratio 

a-\-  c  :h  -\-  e  if  e^  =  ah. 


RATIO,   PROPORTION   AND  VARIATION  27 

^^  a      c      e      ,         ,,    ,    ma? -{- iiac  +  ve^      a? 

4.  If  7  =  ^  =  :^'  show  that  —H)-, — ,-,  ,      /H2  =  T5' 

5.  A  ratio  of  greater  inequality  is  increased  and  one  of 
lesser  inequality  is  diminished  by  taking  from  both  terms 
of  the  ratio  any  quantity  which  is  less  than  each  of  them, 
all  the  numbers  involved  being  positive. 

Q    If  -^  "^^  =  — ,  find  X  without  clearing  of  fractions  {i.e. 
1  —  X      n 
without  cross  multiplication). 

7.  If  -^  =  -1  =  -2  =  -3  =  T'^  prove  that 

^0      ^1      h      h 

K  +  3  Kh  +  3  hK  +  3  h^h}  +  h/ 

8.  If  7  =  -^,  and  a,  5,  c,  d  are  positive  and  in  order  of 

0      d 

magnitude,  prove  that  a-^  d>h  -\-  c. 

9.  If  ax^  +  2  5a;?/  +  cy^  =  0,  find  the  ratio  of  x  to  ?/. 

10.  The  number  of  students  in  Syracuse  University  in  1905 
was  2451;  in  1906  the  number  was  2776.  What  is  the  ratio 
of  the  increase  to  the  number  in  1905  ?  (This  ratio  is  called 
the  rate  of  increase). 

11.  If  a  —  h:c  —  d  =  a-\-h:c  +  d  prove  that  a:h  =  c:d. 

12.  li  a-\-h:c  -\-  d=  ah  :  cd  prove  that  ac  :  bd  =  c  —  a:h  —  d. 

13.  If  a:b  =  c:d,    prove    ab -^  cd   is  a  mean  proportional 

between  a'^-\-c^  and  b^-i-d^. 

J,        J  -,  a^  +  c^      ja  +  cY 

14.  It  a\  b  =  c:  d,  prove  —. ^  =  —r-^ — j-^- 

^  ¥  +  d^      {b  +  df 

15.  If  x  —  y:X=y  —  z:  Y=z  —  x:  Z,  where  x,  y,  z  are 
unequal,  then  X+  Y-\-  Z=0. 

16.  U  a(y +  z)  =  b{z  +  x~)  =  c(x-hy}, 
then 


x—y     _     y  — 'z     _     z  —  x 


c  (a  —  5)      a  (b  —  c)      b{c—  a) 


28 


COLLEGE  ALGEBRA 


VARIATION 

53.  Definition.  If  the  ratio  of  two  quantities  x  and  y  is 
constant  as  x  and  y  take  different  values,  then  x  is  said  to  vary 
directly  as  y. 

If  -  =  ^,  then  x  =  ky  OY  y  =  -x  and  it  is  evident  that  if  x 
y  k^ 

varies  directly  as  y,  then  y  varies  directly  as  x. 

54.  The  statement  that  x  varies  as  y  is  sometimes  written 
xocy^  but  it  must  be  borne  in  mind  that  this  is  merely  a 
method  of  writing  x  =  ky. 

Example.  If  x  —  ky  where  k  is  constant,  what  change 
takes  place  in  y  when  x  becomes  twice  as  large  ? 

k 

55.  Definition.     If  the  product  xy  =  k  ov  x  =  -  where  k  is 

y 

constant,  then  x  is  said  to  vary  inversely  as  y  and  is  sometimes 
written  xcc—. 

y 

If  X  varies  inversely  as  ?/,  then  it  is  obvious  that  y  varies 
inversely  as  x. 

Example.  If  xy  =  k  where  k  is  constant,  what  change 
takes  place  in  y  when  x  becomes  seven  times  as  large? 

Definition.     \i  x  =  kyz  where  k  is  constant,  then  x  is  said  to 

y2ivy  jointly  as  y  and  z. 


GRAPHIC   ILLUSTRATIONS 

56.    Direct  Variation.     In  case 
of  the  straight  line 

y  =  kx  or  ^  =  k, 

X 

the  ordinate  is  k  times  the  abscissa 
for  each  point  on  the  line,  as  has 
been  seen  in  Chapter  I. 


RATIO,   PROPORTIOX   AND   VARIATION 

Y 


29 


*X 


57.  Inverse        Variation. 

Figure   9   is    the    graph    of 

tlie  equation 

I.  ^ 

xy  =  fc  or  x  =  - 

and   is  known  as   an    equi-  ^  \ 
lateral  hyperbola. 

58.  Joint  Variation.  The 
area  of  a  triangle,  which  is 
equal  to  one  half  the  length 
of  the  base  multiplied  by  the 
altitude,  is  an  illustration  of 
joint  variation. 

Denoting  the  area  by  x^  the  altitude  by  y,  and  the  base  by 
3,  we  have  x  =  -yz. 

\i  x  —  -^^  a:  is  said  to  vary  directly  as  i/,  and  inversely  as  z. 


59.    Theorem.     Ifxccy  and  y^z  then  xccz. 
For  X  =  k-^y^ 

y  =  Jc^z, 


(1) 

(2) 


where  k^  and  k^  are  constants.      Substituting  for  y  in    (1) 
from  (2)  we  have  x  =  k  k  z 


or 


X  QC  Z, 


60.  Theorem.  If  xccy  ivhen  z  is  constant^  and  xccz  ivhen 
y  is  constayit^  tJiefi  xac  yz  ivhen  y  and  z  both  vary  together. 

Let  x\  y\  z'  be  simultaneous  values  of  :r,  ?/,  z.  Let  z 
remain  constant  as  y  changes  to  y\  then  x  must  assume  some 
intermediate  value  X  such  that 


X 


y 


(1) 


30  COLLEGE   ALGEBRA 

Now  let  y'  remain  constant  as  z  changes  to  z\  tlien  x  will 
pass  to  the  value  x^  so  that 

-,  =  -,■  (2) 

X       z 

From  (1)  and  (2)  we  have 

X  _  yz 

X       y  z 

x^ 
or  x=  -—  •  yz, 

y  ^ 

where  by  hypothesis  —j—^  is  constant  and  therefore 

y  z 

xozyz. 

61.  EXAMPLES 

1.  If  xaz~,  and  if  a:  =  2  when  ?/ =  3  and  ^=1,  find  x 
when  ^  =  4  and  z  =  5. 

2.  If   xccp-hq,   pccy,    ^oc— ,  and  if  when  ^  =  1,  a;  =18, 

and  when  y  =  2,  x  =  19|,  find  x  when  y  =  11. 

3.  li  x-\-  y  ccx  —  y,  show  that  x'^  -\-  y^cc  xy. 

4.  If  X  varies  directly  as  u  and  inversely  as  v,  when 
uccx(^x  -{-  y^  and  v  Qcxy(x-{-  ?/),  prove  that  :?;  varies  inversely 
as  y. 

5.  If  x^-\-2y'^azxy,  and  a:=l  when  ^  =  1,  show  that  x 
varies  as  y  in  two  ways  and  find  the  ratio  of  x  to  y. 

6.  If  the  square  of  x  varies  as  the  cube  of  y.  and  x=2 
when  y  =  3,  find  the  equation  between  x  and  y. 

7.  If  the  pressure,  volume,  and  temperature  of  a  gas  be 
denoted  by  p,  v,  and  t,  and  if  p  varies  directly  as  (1  H-  at') 
and  inversely  as  v,  and  if  p  =  p^,  v  =  Vq  when  ^  =  0,  find  the 
relation  between  p,  v,  and  t. 


RATIO,   PROPORTION  AND  VARIATION  31 

8.  Two  circular  gold  plates,  each  an  inch  thick  and  hav- 
ing diameters  of  6  and  8  in.  respectively,  are  melted  and 
formed  into  a  single  circular  plate  1  in.  thick.  Find  its 
diameter,  having  given  that  the  area  of  a  circle  varies  as  the 
square  of  its  diameter. 

9.  If  a  body  falls  from  rest,  the  distance  s  passed  over 
(neglecting  the  resistance  of  the  air)  varies  as  the  square  of 
the  time  t.  If  the  body  falls  16  ft.  in  the  first  second,  what 
is  the  relation  between  s  and  t  ? 

10.  If  a  body  falls  from  rest,  the  velocity  v  (neglecting 
the  resistance  of  the  air)  varies  as  the  time  t.  If  at  the  end 
of  two  seconds  the  velocity  is  64  ft.  per  second,  what  is  the 
relation  between  v  and  t? 

11.  What  is  the  velocity  of  the  falling  body  of  Example 
10  at  the  end  of  7  sec.  ? 

12.  If  a  body  falls  as  in  Examples  9  and  10,  prove  that 
the  velocity  v  varies  as  the  square  root  of  the  space  s  passed 
over,  and  find  the  relation  between  v  and  s. 

13.  When  the  body  of  Example  9  has  fallen  625  ft.,  what 
is  its  velocity? 

14.  Prove  that  when  a  body  moves  in  a  straight  line  with 
constant  velocity  the  velocity  v  varies  directly  as  the  space 
s  passed  over  and  inversely  as  the  time  t. 


CHAPTER   IV 
THEORY  OF  QUADRATICS 

62.    If  we  denote  the  roots  of  the  quadratic  equation 


by  rj,  ^2,    then  _  j  4.  V^2  _  4 


ac 


^1  = 


"Aa 


,  —h  —  'Vb^  —  4  ac 

and  ^2  = 

2a 

Adding  the  two  roots  together,  we  have 


_h-\-^'h^-4ae  ,    -b--\/b^-4:ac 
a 
2b  b 


ri  +  r,  = TT. +  2a 


2a  a 

Multiplying  them,  we  have 


_  J  +  V^2  _  4  ^^     _  5  _  V^2  _  4  ^^ 


1'2 


2a  2a 


^  l(^^h)  +  Vb^-4ae\\(-b)-Vb^-4:ac\ 
~  4:a^ 


^(-5)2_(V52_4^g)2 

_  4  <3^6?_  c 
4^2      ^ 

32 


THEORY   OF   QUADRATICS  33 

63.  Dividing  the  quadratic  by  a,  it  takes  the  form 

a         a 

or  using  aS^^  for  r^  +  rg  and  S^  for  r-^r^^  this  equation  takes  the 
form  x^  —  S^x  -\-  iS^  =  0,  from  which  it  is  seen  that  the  sum 
of  the  roots  is  the  coefficient  of  x  with  the  sign  changed, 
and  that  the  product  of  the  roots  is  the  independent  term  in 
this  form  of  the  equation. 

64.  Give  by  inspection  the  sum  and  the  product  of  the 
roots  of  each  of  tlie  following  equations : 

1.  x^-Sx-{-6  =  0.  3.    2a:2  +  4a:-3  =  0. 

2.  x^-h1x-3  =  0.  4.    5x^-7 X +  2  =  0. 

65.  To  form  the  quadratic  equation  whose  roots  are  given. 
From  63  it  is  evident  that  the  quadratic  equation  whose 

roots  are  r^  and  r^  is 

x'^  -  (7\  -\-r^}x-\-  7\r^  =  0, 

which,  by  factoring  the  left  member,  takes  the  form 

(a?  —  r{)  (^x—  r^)  =  0. 

If  then  we  are  given  the  roots  of  a  quadratic  to  form  the 
equation,  we  can  either  take  the  negative  of  the  sum  of  the 
roots  for  the  coefficient  of  x  and  their  product  for  the  inde- 
pendent term,  or  Ave  can  multiply  together  the  factors,  x 
minus  one  root,  and  x  minus  the  other  root,  and  place  the 
product  equal  to  zero. 

66.  Find  in  both  ways  the  equations  whose  roots  are  : 

1.  1,  2.  3.    if  5.    2+ V3,  2- V3. 

2.  -  3,  1 .  4.    3,  -  1 

D 


34  COLLEGE   ALGEBRA 

FACTORING  QUADRATIC   EXPRESSIONS 
67.    Since  every  quadratic  equation 

ax^  -\-hx-\-  c  =  ^ 

can  be  solved  and  written  in  the  form 

a(x—  r^)  Qx  —  r^)  =  0, 

it  is  evident  that  the  factors  of   the    quadratic    expression 
in  the  left-hand  member  of  the  equation  are  aQx—  r^(x—r^. 
This  may  also   be  seen  from  writing  the  expression  ax^ 
-\-hx-\-  c  in  the  form 

a[  x^-{--x-\-  —-  -  — ^  +  - 
a        4  a-^      4  a-      a 


^        V62-4ac\/     ,    h        VJ2_4^^ 
or  a[x  -\- 1 ][x-\-  -; 


2a  2a       J\        2a  2a 


_h-  V52  _  4  ^  A  /         _  J  +  V^/2  _  4  ^^ 

or         a\x x 


2  a  J  \  2  a 

or  a(x  —  r^(x  —  r^, 

68.    It  is  to  be  observed  that  not  only  expressions  of  the 
form  ax^  -\-hx-\-  a  but  also  expressions  of  the  form  ap^  -|-  ap 
+  6',  where  p  is  any  algebraic  expression,  can  be  factored. 
Thus 

(2  x^-^xy-\-4:  iff  -  9  (2  ^2  -  5  2:?/  -f  4^2)  ^2  ^  44  ^4 
=  (2  a;2  -  5  :^^^  +  4  ^2  _  2  ^2)(2  a;2  -  5  a;j/  +  4  ^2  _  7  ^2) 

=  (2 a;2  —  5  rr^  +  2 y2)(2 x^—^xy  —  ^ y'^) 
=  (2x-y^{x-2y)(2x^-  y)(^x -  3  ?/). 


THEORY   OF   QUADRATICS  35 

69.  Since  every  quadratic  equation  in  which  the  coefficient 
of  the  second  power  of  the  unknown  is  unity  can  be  written 
in  the  form  ^^^  _  ^ ^  (^  _  ^^^  ^  q, 

we  may  make  use  of  this  fact  to  solve  quadratic  equations 
whose  left-hand  members  can  readily  be  factored.  For  we 
have  but  to  apply  the  obvious  theorem  that  if  the  product  of 
two  or  more  factors  is  equal  to  zero,  at  least  one  of  the  factors 
must  be  zero,  and  its  converse,  that  if  any  factor  of  a  product 
is  zero  the  product  is  zero,  and  hence  any  value  of  x  which 
makes  any  factor  zero  gives  a  solution  of  the  equation. 

Thus  a;2-2:-12  =  0, 

or  (x-4:){x-\-o)  =  0, 

and  therefore  the  roots  are  4  and  —  3. 

EXAMPLES 

70.  Factor  the  following  : 

I,  2x^-x-S.  2.    2:2  -  24  a:  -  640. 

3.  2(a:H-2)2-T(.r+2)  +  3. 

4.  (a^^Qx-\-Sy-\-S(x^  +  6x  +  S}-\-12, 

5.  (x^  +  x-{-iy-(ix'^-^x  +  l')-2. 

Solve  by  inspection  the  following : 

6.  x'^-{-2x-S5  =  0.  7.    x^-7x-S0  =  0. 

8.  6  2;2-7  2;+2  =  0. 

Solve  by  factoring : 

9.  (:r2-22:+ 3)2-13(2:2- 22^+3) +  22  =  0. 

10.  If  the  expression  2-2  —  3  2:  +  1  has  the  two  values  2  and 

—  3,  find  the  equation  in  x. 

11.  If    the  expression  0^ -\- 2  xy  —  2  y^  is  equal  to  ?/2  and 

—  3  ?/2,  find  the  equation  in  x. 


36  COLLEGE  ALGEBRA 

NATURE  OF  THE  ROOTS  OF  A  QUADRATIC 

71.    It   will   be   seen   from    the    roots    of    the   quadratic 
equation 


_J4.V62- 

-  4ac 

2a 

_  h  -  V^>2  _ 

-  -iac 

5 

2a 
that  they  are 

(1)  real  and  distinct  if    5^  >  4  ac^ 

(2)  real  and  equal  if        5^  _  4  ^^^ 

(3)  imaginary  if  5^  <  4  ac. 

In  other  words,  the  conditions  that  the  roots  are  real  and 

distinct,  real  and  equal,  or  imaginary,  are  that  }p'=^\ao 
respectively. 

The  roots  will  be  rational  or  irrational  according  as  5^  —  4  ac 
is  or  is  not  a  perfect  square. 

72.    If  (?  =  0,  the  two  roots  become 

or  0, 


2a 

-h-VP  h 

and or 

2a  a 

In  other  words  the  condition  that  one  root  should  be  zero 
is  that  c  should  be  zero. 

If  (?  =  0,  the  equation  takes  the  form 

x(^ax  +  5)  =  0, 

showing  that  a;  is  a  factor  of  the  left-hand  member. 

The  condition  that  two  roots  should  be  zero  is  that  5  =  0  as 
well  as  c  =  0,  as  may  be  seen  from  the  roots  themselves. 


THEORY  OF   QUADRATICS  3T 

Under  these  two  conditions  the  equation  takes  the  form 

ax^  —  0, 

showing  that  oc^  is  a  factor  of  the  left-hand  side. 

73.    If  we  multiply  both  numerator  and  denominator   of 
the  root 

2a 


by  —  5  —  V52  —  4  ac^  we  have 


_   ^  +  V^>2  _  4  ^^  _   ^   _  V52  _   4  ^^  ^         (   _   5)2  _    (^,2  _    4  ^g^) 

2  6' 


5  _  V52  -  4 


ac 


Similarly  multiplying  the  numerator  and  the  denominator 
of  the  root 

—  h  —  V52  —  ^ac 
'  2^ 


by  —h-\-  -\^b'^  —  4  ac^  we  have 

2c 


_  5  4.  V62  -  4  a(? 


If  now  a  =  0,  the  roots  become 


2c  c         .2c 

,  or  —  -,  and  — -,  or  00  . 


-2V  b  0 

If  both  a  and  h  are  equal  to  zero,  then  both  roots  are 
infinite. 

In  other  words,  the  condition  that  one  root  is  infinite  is  that 
a  =  0,  and  the  conditions  that  two  roots  are  infinite  are  that 
a  =  0  and  that  5  =  0. 


38  COLLEGE   ALGEBRA 

74.  If  5  =  0,  while  a  and  c  are  not  zero,  the  two  roots 
become  equal  in  numerical  value  but  opposite  in  sign.  If 
a=  c^  the  roots  are  the  reciprocals  of  each  other,  for  then 

r.r^  =  -  =  1. 
a 

These  conditions  for  infinite  roots  might  have  been  ob- 
tained by  writing  x  =  -  and  applying  the  conditions  for  zero 

roots  for  the  quadratic  in  ?/. 

75.  The  expression  5^  —  4  ac  is  called  the  discriminant  oi 
the  quadratic. 

Statements  analogous  to  those  which  have  been  made  con- 
cerning the  nature  of  the  roots  of  a  quadratic  equation  can 
be  made  concerning  the  nature  of  the  factors  of  a  quadratic 
expression. 

76.  The  roots  of  an  equation /(a^)  =  0  represent  geometri- 
cally the  points  on  the  2;- axis  where  the  curve  ?/  =f(x) 
meets  it. 

For  example,  —  1  and  3,  the  roots  of  the  equation 

2;2-2a;-3  =  0, 

represent    the    points    on     the     :r-axis    where    the     curve 
y  =  (x-\-  'V)(x  —  3)  meets  it. 

It  is  seen  that  to  obtain  the  coordinates  of  the  points  is 
the  same  as  to  solve  the  two  equations 

y  =  x^  —  2  X  —  ?>^ 

and  y  =  ^' 

77.  The  geometrical  meaning  of  the  foregoing  conditions 
as  to  the  nature  of  the  roots  of  a  quadratic  may  be  illus- 
trated by  the  following  example  : 

x^—  S  X  i-p. 


THEORY   OF   QUADRATICS 


39 


Here  h^  —  4:ac  =  9  —  4p  and  according  as  9  —  4 p  =  0,  i.e. 


< 


according  asjt?^^  the  roots  are  real  and  distinct,  real  and 

equal,  or  imaginary.    I 

As  we  have  just  seen,  the  roots  of  the  equation  represent 
the  points  on  the  axis  of  x  where  the  curve 


meets  the  line 


i/  =  x^— Sx-{-p 
y  =  0. 


Since  any  increase  in  the  value  of  p  lengthens  all  the 
ordinates  by  the  amount  of  that  increase,  it  elevates  the 
whole  curve  with  respect  to  the  2;-axis. 

If  now  JO  =  2,  the  curve  cuts  the  axis  of  x  at  the  points 
x=l  and  a;  =  2.  This  is  the  case  of  real  and  distinct  roots, 
Fig.  10. 


>x 


Fig.  10. 


If  p  is  increased  to  |,  the  curve  is  elevated  and  the  two 
points  in  which  it  meets  the  axis  of  x  come  to  coincide  at 
a:=|,  and  the  2:-axis  is  tangent  to  the  curve.  This  is  the 
case  of  equal  roots.  Fig.  11. 


40 


COLLEGE   ALGEBRA 


>X 


Fig.  11. 


If  p  is  given  a  value  greater  than  |,  the  curve  is  so 
elevated  that  it  no  longer  cuts  the  axis  of  x.  This  is  the 
case  of  imaginary  roots,  Fig.  12. 


>x 


Fig.  12. 


THEORY   OF   QUADRATICS  41 

78.    Theorem.     Every  quadratic  equation^ 

ax^  -{-hx-{-  c=0^  where  «  ^  0, 

has  two  and  only  tivo  roots. 

It  has  already  been  shown  in  62  that  it  has  two  roots,  viz. 


'^i=-^A 


h    .  V52_4 


ae 


2  a  2  a 


_        h       V52  —  4:  ac 
^2~  ~2~a  2"a 

If  possible  suppose  that  it  has  another  root  rg,  different 
from  r^  and  r^'     Then  we  should  have 

ar^^  -{-br^  +  c  =  0,  (1) 

ar^^  +  5r2  +  ^  =  0,  (2) 

argS  +  hr^  +  c=0.  (3) 

Subtracting  (1)  from  (2)  gives 

or  a(r2  +  r^)  H- 6  =  0.  (4) 

Similarly  subtracting  (1)  from  (3)  gives 

«(^3  +  ^i)  +  6  =  0.  (5) 

Subtracting  (0)  from  (4)  gives 

which  is  impossible,  since  a=^0  and  r^  —  r^^O  by  hypothesis. 
The  supposition,  therefore,  that  the  equation  has  a  third 
root  different  from  r^  and  r^  is  false,  and  the  equation  has 
two  and  only  two  roots. 


42  COLLEGE   ALGEBRA 

Another   and   shorter   proof    of    this   proposition   is   the 
following  : 

If  r^  and  r^  are  two  roots  of  the  quadratic  equation 

ax^  +  bx-\-  c=  0, 

we  have  seen  in  67  that  it  may  be  written  in  the  form 

a(^x  —  r^')(x  —  r^)  =  0. 

If  now  rg  is  a  root,  it  follows  that 

and  therefore  either  r^  —  r^  =  0,  or  r^—  r^  =  0,  and  hence  r^ 
is  not  different  from  r^  or  r^. 

79.  EXAMPLES 

1.  Find  the  sum  and  product  of  the  roots  of  the  equation 
(a  +  2  5)a;2 -  ^bcx-{- c-2  d  =  0. 

2.  Find  for  what  values  of  \  the  equation 

ax^  —  2  b\x  +  3  (?  —  X  =  0  has  equal  roots. 

3.  For  what  values  of  m  are  the  roots  of 

mx'^  H-  (a  4-  2  m)x  +  (3  c  +  m)  =  0  imaginary? 

4.  If  the  equation  5  a;  —  10  =  0  is  regarded  as  a  quadratic 
equation,  what  are  its  roots  ? 

5.  For  what  values  of  X  is  a  root  of 

(a2 -  X2)^2 -.2a\x  +  \'^-\-Sa^=0  infinite ? 
What  is  the  other  root  ? 

6.  What    is    the    nature    of    the   roots   of    the   equation 
Sx^-12x+5  =  0? 

7.  What   is   the    nature    of    the    roots    of    the    equation 
2a;2-32;  +  2  =  0? 


THEORY   OF    QUADRATICS  43 

8.  For  what  values  of  a  and  h  are  the  roots  of  the  equa- 
tion (3a  +  5-2)a;2+(2a-3^-l)a;  +  a  +  2^  =  0  both  in- 
finite ? 

9.  For  what  values  of  a  and  h  are  both  roots  of  the  equa- 
tion (a  -f-  2  5)  :z;2  —  (3  a  —  5  ^  +  5)  a;  4-  («  —  2  5  +  4)  =  0  equal 
to  zero  ? 

10.  When  a  is  any  square  number,  prove  that  the  roots 
of  (a  —  1)  a;^  —  2  (a  —  2)  a;  H-  «  —  4  =  0  are  rational. 

11.  What  is  the  nature  of  the  factors  of  2^:^— 72;  +  2? 

12.  What  is  the  nature  of  the  factors  of 

(2^2  _  a:  +  l)2_4(a;2  -  a;  +  1)  +  3? 

13.  The  sura  of  two  numbers  is  30  and  their  product  is 
221.     What  are  the  numbers  ? 

14.  The  expression  2  a:^  —  3  a;  —  3  has  two  values,  the  sum 
of  these  values  is  —  5  and  their  product  is  4.     Find  x. 

15.  The  sum  of  the  two  values  of  a^'^  —  3  a;  + 1  is  —  1  and 
their  product  is  —  12.    What  is  the  nature  of  the  values  of  a;  ? 

16.  Determine  x  so  that  the  sum  of  the  two  values  of  the 
expression  x^ -\- x—1  may  be  5  and  their  product  6. 


CHAPTER   V 

FACTOR,    IDENTITY,   AND    REMAINDER   THEOREMS 

80.  Factor  Theorem.  If  P  represents  the  polynomial  of  the 
nth  degree  ^  „_i  n-2  _i_  .    «-3  ,  . 

which  vanishes  when  a;  =  « ,  then  x—  a  is  a  factor  of  P, 

Since    a^a""  +  a^«"-i  +  a^a'^''^  +  a^a'^~^  +  . . .  +  ^^^  =  0 

by  hypothesis,  we  may  subtract  this  expression  from  P  with- 
out altering  its  value. 
We  have 

P  =  «^2:"  -t-  a-^x^^''^  +  «2^""^  +  •  •  •  +  a„_  jO;  +  a,^ 

—  (^0^^"  +  a^aJ"-^  +  a^a''-'^  + \-  a,,_^ci  +  a^) 

=  ao (x^  -  a'')  +  ^1  (x""-^  -  «"-!)  +  «2(^""^  -  «''"^)  +  •  •  • 

and  as  every  term  of  this  expression  is  divisible  by  x  —  «,  it 
follows  that  P  is  divisible  by  a:  —  «  according  to  353.  This 
theorem  is  known  as  the  factor  theorem. 

Example.  2x^-\-x  —  1  vanishes  when  2:=  — 1,  for  it 
then  becomes  2  —  1  —  1  =  0.  Hence  x-\-l  must  be  a  factor. 
Similarly,  the  same  expression  vanishes  for  x  =  ^.  Hence 
a;  —  J  is  also  a  factor.     In  fact 

2  a;2  +  a;  -  1  =  2  (a;  -  1)0-^  -}- 1). 
44 


FACTOR,   IDENTITY,   AND   REMAINDER   THEOREMS     45 

81.  Number  of  Roots  of  an  Equation.  The  proof  of  the 
fact  that  every  equation  has  a  root  is  not  simple  and  may 
be  found  in  Chapter  XIX. 

Assuming  this  fact  or,  what  is  the  same  thing,  that  every 
polynomial  vanishes  for  at  least  one  value  of  the  unknown, 
it  is  easy  to  show  that  every  equation  of  the  nth  degree  has  n 
roots. 

If  «j  is  a  root  of  the  equation 

fn(x)  =a^pf^  +  a-^x"~'^  +  a^x"'-'^  -f  ...  -f  a,,  =  0,      where     ^q^^O, 

then  by  the  factor  theorem  x  —  «j  is  a  factor  of  the  left-hand 
member,  so  that  it  vnd^y  be  written 

aJx-+^x--^+---^^A=a,(x-a^Y,_^{x\ 
\         a^  a^j 

where /„_-^ (a;)  is  a  function  of  the  {n  —  l)th  degree. 
By  the  same  id^oX  f n_^{x)    has  a  factor  x—  a^,  so  that 

fn  (^)  =  S  (^  -  «l)(^'  -  «2)/«-2(^)  • 

Again 

fn  W  =  «o  (^  -  '^l)  (^  -  «2)  C^  -  «3)/«  -3  C^)' 

and  finally 

fn  (^)  =  ^o(^  -  "l)  (^  -  «2)  (^  -  '"'3)  '-'{X-  «.)  • 

Therefore /„  (2:)  vanishes  for  the  n  values,  «i,  «2'  H'>  "*'  "»• 
This  polynomial  cannot  vanish  for  more  than  n  values,  for 

if  possible  let  it  vanish  for  x  =  P  where  yS  is  different  from 

each  a. 

Then  a^  (^  -  a^)  (^  -  a^)  •  •  •  (^  -  «„)  =  0, 

which  cannot  be  true  unless  some  one  of  the  factors  yS  —  (x-k  =  ^^ 
or  /8  =  a^.,  which,  by  hypothesis,  is  not  the  case.  The  theorem 
of  78  is  a  special  case  of  this. 


46  COLLEGE   ALGEBRA 

82.  Identity  Theorem.  If  a  polynomial  of  the  nth  degree 
vanishes  for  more  than  n  different  values  of  x^  the  coeff  dents  of 
every  power  of  x  vanish. 

For  if  a^  ^  0,  the  polynomial  cannot  vanish  for  more  than 
n  different  values  by  the  preceding  theorem,  but  by  hypothe- 
sis  it   does  vanish  for   more  than  n  values  and  therefore 

"0=0- 

Our  polynomial  is  now  of  the  (?^  — l)th  degree  and 
vanishes  for  more  than  n—  1  values  of  x  and  therefore  a-^  =  0. 

Similarly   ^2  =  ^3  =  ^4  =  •  •  •  =  ^«  =  0?    and   the   polynomial 

0x''-i-0x''-^  +  0x''-^-{ hOa^  +  0. 

This  is  seen  to  be  identically  zero  or  to  vanish  for  all  values 
of  X. 

Thus  it  is  seen  that  if  a  polynomial  of  the  nth  degree  in 
X  vanishes  for  more  than  n  values  of  x^  there  are  three 
equivalent  ways  of  stating  the  conclusion:  1.  The  coeffi- 
cients of  the  various  powers  of  x  vanish.  2.  The  polynomial 
vanishes  for  all  values  of  x.  3.  The  j)olynomial  vanishes 
identically.     This  theorem  is  called  the  identity  theorem. 

83.  Identity  of  Two  Polynomials.  If  two  polynomials  of 
the  nth  degree  are  equal  to  each  other  for  more  than  n  values 
of  x^  the  coefficients  of  the  corresponding  powers  in  the  two 
polynomials  are  equal. 

Let  the  two  polynomials  be 

a^x^  +  a^x'^~'^  -\-  a^x^^"^  -\-  -•'  +  cin 
and  h^x^  -^  h-^x''-'^  +  h^x''-'^  -\ \-  h^. 

Then  if  these  are  equal  for  more  than  n  values,  it  follows 
x^ (^0  -  5o)  +  2;«-i (aj  -  5i)  +  •  •  •  4-  {a,  -  h,) 


FACTOR,  IDENTITY,   AND  REMAINDER   THEOREMS     47 
vanishes  for  more  than  n  values,  and  therefore 


a 


(j-5o=0,  ai-^i  =  0,  .-.,  a„-hn  =  0,  (82) 

or  «o  =  ^0,  «i  =  ^1,  •  •  •,  cin  =  h^. 

Let  the  student  state  two  other  forms  for  the  conclusion 
of  this  theorem. 

Example.  Find  the  condition  that  a^x^  +  2  a^x  +  ag  may- 
be a  perfect  square. 

If  the  expression  is  to  be  a  perfect  square,  the  square  of 
ax+  yS  say,  we  must  have 

a^py^  +  2  a^x  -\-  a<^=  {iix  +  ^y- 

therefore  o^  =  a^,  «/3  =  a^,  ^  =  a^j, 

whence  d?'^  =  a^a^,  also  (^a/sy  =  a^/3-  =  a^, 

and  therefore  a^a^  =  af^  or  -5  =  -^  • 

The  student  will  see  that  this  agrees  with  the  conditions 
of  71  that  4  a^2  _  4  ^^^^  _  q,  or  a-^  =  a^^g- 

84.  The  foregoing  gives  us  a  convenient  means  of  finding 
by  trial  the  factors  of  some  polynomials. 

Thus,  in  p  =  ^3  _  6  ^2_,_  11  ^_  6, 

if  we  put  x  =  l  we  have 

l_6  +  ll-6=0, 

and  therefore  rr  —  1  is  a  factor. 
If  we  put  x=2  we  have 

8  -  24  +  22  -  6  ==  0, 

and  therefore  a;  —  2  is  a  factor. 


48  COLLEGE   ALGEBRA 

Again,  if  we  put  x=  S,  we  have 

27  -  54  +  33  -  6  =  0, 

and  therefore  a^  —  3  is  a  factor. 

Since  P  is  of  the  third  degree  in  x  there  can  be  no  other 
factor  involving  x,  and  if  there  is  any  other  factor,  it  must 
be  numericah     Denoting  it  by  iV,  we  have 

a^-6x^-hllx-6  =  N(x-l}(x-2)(x-S), 

and  since  this  is  an  identity  the  coefficients  of  like  powers 

of  X  on  both  sides  must  be  equal.     Equating  the  coefficients 

of  a^  we  have 

l=iV. 

Therefore         P  =  (x -l)(ix  -  2)(x -  3). 

85.    Remainder  Theorem.     If  a  polynomial 

P  =/„  (x)  =  a^x^  +  a-^x"^'^  +  a^x""'^  +•••+«« 

he  divided  hy  x  —  a,  the  remainder  will  be  the  result  obtained  by 
substitutifig  a  for  x  in  P. 

Let  Q  be  the  quotient  which  Ave  obtain  on  dividing  P  by 

X—  a  until  we  get  a  remainder  P,  which  does  not  contain  x, 

so  that  ^       .  . 

P=Mx}  =  Q(x-a)-^E, 

which  is  an  identity,  true  for  all  values  of  x,  since  it  is  true 
for  more  than  n  values  of  x  different  from  a,  and  hence  true 
for  x  =  a. 

Placing  x=  a,  we  have 

/«(«)  =  ^o^""  +  ^i""""^  +  ^2^""^  +.-.  +  «,,  =  72. 
Tliis  is  called  the  remainder  theorem. 


FACTOR,   IDENTITY,   AND   REMAINDER   THEOREMS     49 

86.  EXAMPLES 

Factor : 

1.  :c3  +  5a;2- 9a:-45.  3.    r^ -\-2  x^  -  2Sx- 60, 

2.  x^-a^-S9x^-^2^x-\-lS0.       4.    x^- ^x"^  - x+ 6. 

Find  the  remainder  when  : 

5.  a^  —  Sx^—5x-\-lis  divided  by  a:  —  2. 

6.  xi^  —  2x'^-\-Sx—o  is  divided  by  2^  +  4. 

7.  Ux^-^x^+2x-l  =  ax'^-hC2a  +  b)a^-\-(^b-c)x'^ 

-i-(c-2d)x-d-\-2e, 
find  the  values  of  a,  5,  c,  d,  e, 

8.  Prove  without  assuming  the  result,  that  -O  =  -i  =  -2  is 

the  condition  that  a^oi^  H-  3  a-^x^  +  3  a^x  4-  a^  may  be  a  perfect 
cube. 

9.  Find  the  condition  that  2:^  +  3  Hx  +  (r  may  contain  a 
square  factor  (x  +  «)2 . 

Suggestion.     Assume  x^  -\-?>  Hx-\-  G  =  {x-\-  ay-^x  +  (i). 

10.  Show  that  the  values  of  the  5's  must  all  be  unity  in 
order  that 

Factor : 

11.  x^-Qs^  +  ^x^+12x-m. 

12.  x^-6x^-\-\lx-Q. 

13.  :?:*+22:3_3^_4^4.4, 

14.  2;4-10.T3  +  35a^- 502-4-24. 

15.  a%-\-  ah'^  -\-  IP-c  +  ?>c2  +  c^a  +  a^c-i-2  ahe. 

16.  a^h-\-ab''-'-{-h'^c  +  hc^^c'^a-\-a^c-{-^ahe. 


CHAPTER   VI 

COMMENSURABLE,   INCOMMENSURABLE,  AND  IMAGINARY 

NUMBERS 

87.  Definitions.  The  work  in  elementary  algebra  involves 
positive  and  negative  integers,  zero,  and  positive  and  negative 
fractions.  These  numbers  are  all  known  as  commensurable 
numbers. 

Suppose  we  are  given  x^  =  w,  where  m  is  any  commensura- 
ble number.  Then  x  is  called  the  square  root  of  w,  and  may 
be  defined  as  that  number  which  when  multiplied  by  itself 
will  produce  m.  It  is  represented  symbolically  by  x  =  Vm! 
Three  cases  will  arise : 

First,  suppose  that  m  is  a  perfect  square  of  some  commen- 
surable number,  so  that  m  =  t^.  Then  our  definition  of  x 
will  permit  it  to  have  either  of  the  values,  ^  or  —  ^.  Thus 
it  will  be  seen  that  the  square  root  of  m  has  two  values. 
For  the  sake  of  simplicity,  unless  there  is  some  particular 
reason  to  the  contrary,  the  symbol  Vm  is  taken  to  represent 
the  positive  root,  so  that  the  two  values  of  x  are  represented 
by  Vm  and  —  Vm,  or  by  the  combined  symbol  ±  Vm. 

Second,  suppose  that  m  is  positive,  but  not  a  perfect  square, 
as,  for  instance,  2.  Then  no  commensurable  number  can 
possibly  represent  V2,  though  by  taking  a  sufficient  number 
of  decimal  places,  a  number  can  be  found  which  will  satisfy 
the  conditions  within  any  prescribed  degree  of  accuracy  of 
approximation. 

50 


IMAGINARY  NUMBERS  51 

For,  if  possible,  suppose  that  V2,  which  is  equal  to 
1.4142135- ••,  as  may  be  found  by  trial,  is  a  number  whose 
decimal  part  terminates.  Then,  squaring  both  sides,  we 
have  an  impossibility,  as  the  right-hand  member  cannot 
be  2.  Hence  the  supposition  that  the  expression  V2  is  a 
terminating  decimal  is  false.  Neither  can  V2  be  expressed 
by  a  commensurable  fraction.     For,  if  possible,  suppose  that 

V2=-,  where  -  is  a  commensurable  fraction  reduced  to  its 

0  0 

lowest  terms.     Then,  squaring  both  sides,  we  again  have  an 
impossibility,  as  —   is  a  commensurable  fraction  reduced  to 

its  lowest  terms  and  cannot  be  equal  to  the  integer  2. 

Such  a  number,  whose  value  cannot  be  exactly  expressed, 
either  by  an  integer  or  by  a  fraction,  but  which  can  be  ap- 
proximated within  any  prescribed  degree  of  accuracy  by  using 
a  sufficient  number  of  decimal  places,  is  called  an  incommen- 
surable number. 

Third,  suppose  that  m  is  a  negative  number,  as,  for  instance, 
—  3,  or  —  4.  The  principles  of  elementary  algebra  show  that 
no  positive  or  negative  number  when  multiplied  by  itself  can 
possibly  produce  a  negative  number.  Hence  if  we  are  to 
represent  a  true  solution  of  the  problem,  we  must  introduce 
a  new  kind  of  number.  Thus  we  define  V—  n^  Avhere  n  is 
intrinsically  positive,  as  the  number  such  tliat  (±V  — n)^ 
=  —  n,  and  such  a  number  is  called  a  pure  imaginary.  It 
obeys  all  the  laws  of  algebra,  and  its  properties  will  be  ex- 
plained in  the  last  division  of  the  present  chapter. 

In  contrast  with  the  pure  imaginaries,  those  numbers  which 
have  been  defined  as  commensurable  and  incommensurable 
are  called  real  numbers. 

A  combination  of  real  and  imaginary  numbers  in  the  form 
a±V— 6,  where  a  is  real  and  b  is  intrinsically  positive,  is 


52  COLLEGE   ALGEBRA 

called  a  complex  iiiimber.  The  properties  of  complex  num- 
bers will  be  discussed  in  connection  with  those  of  pure 
imaginary  numbers. 

THEORY   OF  INDICES 

88.  The  proof  of  the  fundamental  theorems  of  the  theory 
of  indices  for  positive  integral  exponents  *  are  given  in 
elementary  algebra,  but  for  the  sake  of  completeness  the 
following  are  reproduced  here: 

If  m  and  n  are  positive  integers,  and  x  is  any  commensu- 
rable number,         ,^         .,.,..       -i-         -c    ^ 

'       x^^  =  X  X  X  X  X  X  •••  to  m  tactors, 

and  x^  =  xxxxxX'--to  n  factors. 

Therefore      x"^  xx'^  =  (x  x  x  x  x  x  •  •  •  torn  factors) 

x(xxxxxx---ton  factors) 
=^  X  XX  X  X  X  ■••  to  (m  +  7i)  factors 

Applying  the  above  principle  twice,  we  have 

^7«i  X  2:'"2  X  a;"'3  =  (af^^  X  x"^^~)  X  x^^ 

Continuing  the  process,  we  have 

If,  now,  m-^  =  m^  =  m^=  -"  =  771^  =  ^1, 

this  becomes      x""  x  x"^  x  x'"  x  •••  to  n  factors, 
or,  (x"y'  =  x""^ 

Corollary.     Likewise  (^x'^y  =  a^'""  =  x"""". 
Therefore  (x'^y  =  (x'^y. 

*  In  this  book  the  terms  index  and  exponent  are  used  interchangeably. 


IMAGINARY  NUMBERS  53 

If  X  and  y  are  two  commensurable  numbers,  and  ri  is  a 
positive  integer, 

rjiaiyn  =z(xv.xy.xy.--'ion  factors) 

x(t/  X  y  X  y  X  •■'  to  n  factors) 
=  (xy)  X  (xy)  X  (xy)  x  •••  to  n  factors 

Finally,  if  m  and  n  are  positive  integers,  and  m  >  n^ 

„      rr'"      x  X  X  X  X  X  •••torn  factors 
x'^      X  XX  XX  X  "•  to  n  factors 

=  X  X  X  XX  X  •••  to  (m  —  n~)  factors 

^m—n 

89.  In  88  it  is  assumed  that  the  exponent  n  is  a  positive 
integer.  In  the  remainder  of  this  chapter,  the  meaning  of 
x"^  will  be  extended  so  as  to  allow  n  to  have  any  commensu- 
rable value.  Consistency  requires  that  the  extension  shall 
be  made  in  such  a  way  that  the  laws  for  positive  integral 
exponents,  as  heretofore  developed,  and  the  other  funda- 
mental laws  of  algebra  shall  be  obeyed.  It  will  be  shown 
that  if  all  the  new  exponents  are  defined  by  the  requirement 
that  they  must  satisfy  the  law  x"^x'^  =  a:"'^",  for  positive 
integral  exponents,  then  they  will  satisfy  all  the  other  laws 

for  positive  integral  exponents. 
1 

90.  Since  a;%  where  n  is  a  positive  integer,  must  obey  the 

same   law   as   2:%    where   71    is   an    integer,    we    must    have 

111  1 

x'^  •  a;"  •  a;"  •••  to  ^  factors,  or  (2:")",  equal  to 

-+-+-+•••  to  n  terms 

From  (a;")  =  x^  we  have  x^  =  Va;,  by  taking  the  nth  root  of 
each  side. 


54  COLLEGE   ALGEBRA 

11  1 

Suppose  x{^x^  '  ■  ■  x,^  =  ci", 

1 

and  (^1^2  •  •  *  ^'n)"  ~  ^' 


Raising  each  side  of  both   equations  to  the  Tith  power,  we 


get  x,X. 


Therefore  a"  =  5%  or  a  =  6, 

111  1 

Raising  both  sides  of  this  to  the  pih.  power,  we  have 

p    p  p  p 

♦*'l  •*'2    " '       »t    —  V'*'l'*^2         *^/ny    • 

Therefore  positive  fractional  indices  obey  the  law 

the  same  as  a  positive  integer  n. 

91 .    In  90  suppose  that  each  of  the  m  factors  x^x^  •••  Xj^  is 
equal  to  x,  then,  i  l        ^ 

(cc")""  =  (x'^y  =  x^, 
1       _ 

or,  since  x"^  =  "v/a:, 

m 

therefore  ( V^;)'"  =  V:r'"  =  a:"^, 

which  shows  that  the  m\h  power  of  the  n\X\  root  of  a  number 
is  equal  to  the  n\h  root  of  the  mth  power  of  that  number, 
and  that  both  are  equal  to  that  number  with  a  fractional 

ryyh 

exponent  — • 

n  11 

Again,  let  {pc^y  =  a, 

1  1 
and  {xT'y'  =  6. 


IMAGINARY   NUMBERS 


5b 


Then, 

and 

Therefore 


1 

x""  =  6%  and  x  =  b'" 

^mn  ^  f^mn^      OT  a  =  b, 


Again, 


Therefore 


which  shows  that  the  nth  root  of  the  mth  root  of  a  number 
is  the  same  as  the  mth.  root  of  the  nth.  root  of  the  same 
number. 

x'i  =  (x'ly. 

(^p  and  q  being  positive  integers) 

p   m  11 

(m  and  n  being  positive  integers) 
1  1 

pm 

1    1 

1    1 

=  [\(x"y}'^y 

m   p 

=  (x^y. 

It  is  therefore  seen  that  positive  fractional  indices  obey 

tne  law  fry,m\n  ^  r^'n\m  _.  ^mn 

the  same  as  positive  integers  m  and  n. 

Since  the  zero  exponent  must  obey  the  law,  x"^x^  =  a:"^"*"",  the 
same  as  positive  exponents  m  and  n, 
we  have  '**"*  ■  '^^  —  ryju+o 


X" 


X^  =  X" 


X 


=  X" 


:rO  =  — =  1. 

X'" 


Therefore 

Likewise  negative  exponents  must  obey  this  same  law. 


56  COLLEGE   ALGEBRA 

Therefore  a;™  •  x~"^  =  a;'"-'"  =  x^  =  1 . 

Hence  x~"^  =  — . 

Therefore  any  number  (zero  sometimes  excepted)  raised  to 
the  zero  power  is  equal  to  unity,  and  any  number  with  a 
negative  exponent  is  equal  to  the  reciprocal  of  that  number 
with  exponent  positive. 

The  negative  exponent  also  obeys  the  law  (2:'")^=  (a;")'"  =  a:^"". 

For  (a:-"0"  =  ( — Y  =  —  =  ^"""'  =  -^—  =  (x"")  -™. 

Also      (x'^y  =  (  —  )  "  =  -i-  =  x"'""  =  — - —  =  (2:-")-^. 

Likewise  the  negative  exponent  obeys  the  law 
For 

Hence  we  have  shown  that  by  requiring  all  new  exponents 
to  satisfy  the  law  x'^'x^  =  x"^'^^,  it  follows  as  a  consequence  that 
they  also  satisfy  the  laws 

and  rri":r/  •••  xj  =  (^x^x^  •••  x„,y. 

92.  EXAMPLES 

Express  with  radical  signs : 

2     \  3      5       2.  ?    r 

1.    a^b".  2.    rri^n^p^.  3.    xiy^ 

Express  with  fractional  exponents: 

4.     ■\/a^W.  5.     V^\/P^^.  6.     V^V/. 


IMAGINARY  NUMBERS  57 

Express  with  positive  exponents  : 
7.    7?y~^z~^.  8.    a~%\c~^,  9.    vrhi'^. 

Express  without  denominators : 

10       ^^^^  11      m^^n^  8a 


a^'¥c' 


Find  the  values  of  the  following  : 

13.  (a^)6.  15.    (m"%^)i^  17.    (8^)5. 

14.  (0:^^)^.         16.    (24^)1  18.    (-729)1 

Multiply  the  following : 

19.  5  a;^  —  a:^  +  7  by  2  a:^  +  3  ^3  _  4, 

20.  2  r?:*  _|_  5  ^3  _  3  ^-2  +  7  by  3  ^-3  _^  7  ^-2  _  n, 

21.  3  c^h^  -  7  a^5^  +  a-W  by  2  a^r^  -  3  a%~^. 

Divide  the  following : 

22.  2:^  +  372:^-700:^  +  50  by  a;^- 2  r?;^  + 10. 

23.  8a-^^-8a^  +  5«=''^-3«-3'^  by  5««- 3a-^ 

24.  5  6^  -  6  5  -  4  -  4  5*  -  5  jHy  5^  -  2  6i 

RADICALS 

93.  Definitions.  Any  number  to  which  a  radical  sign  is 
attached  is  called  a  radical ;  as  V2,  V5,  V9. 

If  m  is  a  commensurable  number  which  is  not  a  perfect 
wth  power,  -\/m  is  incommensurable,  87.  It  is  then  called 
a  %urd.  A  %urd  is  thus  defined  as  an  incommensurable  root 
of  a  commensurable  number.  Thus  V2  =  1.4142- ••  and  can- 
not be  exactly  expressed,  however  far  the  calculation  is 
carried  out. 


58  COLLEGE   ALGEBRA 

It  will  be  observed  that  all  surds  are  radicals,  but  not  all 
radicals  are  surds.  Thus  V9  is  a  radical,  but  not  a  surd. 
V^  is  or  is  not  a  surd  according  as  a  is  or  is  not  a  perfect 
square,  but  in  algebra  all  such  expressions  as  Va,  VS,  etc., 
are  considered  as  surds. 

Any  number  which  can  be  expressed  without  involving 
surds  is  said  to  be  rational. 

All  surds  or  numbers  which  cannot  be  expressed  without 
involving  surds  are  said  to  be  irrational. 

The  order  of  a  surd  is  the  index  of  the  root  which  it  repre- 
sents. Thus  a  surd  which  is  a  square  root  is  of  the  second 
order,  and  one  which  is  an  nth.  root  is  of  the  nth  order. 

It  has  been  shown  that  surds  can  be  written  as  fractional 
powers  of  numbers. 

A  surd  or  radical  which  has  a  rational  coefficient  is  called 
a  mixed  surd  or  mixed  radical. 

A  surd  or  radical  which  has  no  rational  coefficient  except 
unity  is  called  an  entire  surd  or  entire  radical. 

94.  Three  types  of  surds  and  radicals  demand  special 
attention : 

1.  Those  in  which  the  expression  affected  by  the  radical 
has  a  factor  which  is  a  power  of  the  same  degree  as  the  index 
of  the  radical ;  as  V250,  -\/a^h. 

2.  Those  in  which  the  expression  affected  by  the  radical 
is  a  power  whose  index  is  a  factor  of  the  index  of  the  radical; 
as  V8. 

3.  Those  in  which  the  expression  affected  by  the  radical 
is  a  fraction  ;  as  V|. 

95.  Since  it  has  been  shown  in  90  that  surds  and  radicals 
can  be  expressed  without  the  radical  sign  by  means  of  frac- 
tional exponents,  the  laws  of  exponents  furnish  two  princi- 
ples which  are  of  use  in  the  reduction  of  surds  from  one  form 
to  another. 


IMAGINARY   NUMBERS  59 

1.  ^~ab  =  Qahy  =  a^f^=-^a-</h, 

m  1 

2.  "'■VaJ^'  =  a^'  =  a^'  =  </a, 

These  principles  permit  a  transformation  of  the  three  types 
of  94.     Thus, 

■^250  =  ^125^2  =  VJ2b  .  V2  =  5^. 

-^8  =  ^23  =  V2. 

Since  the  denominator  in  the  last  case  must  be  a  perfect 
square,  the  first  step  must  be  to  multiply  both  numerator 
and  denominator  by  3. 

These  reductions  show  that  it  is  permissible  to  define  a 
surd  as  in  its  simplest  form  when  the  expression  affected  by 
the  radical  is  integral  and  as  small  as  possible. 

96.  EXAMPLES 

Reduce  the  following  to  their  simplest  form: 


1.    Vl2.  5.    ^/32.  9.   -Vimba^y^. 


2.    Vl8.  6.    V4a.  10.    V«2  -  2  a6  +  ^2. 

3   /TT^ 


3.    V32.  7.    V27  a%.  11.   V2  x^  +  4:xy-{-'l  y^. 


4.    V24.  8.    V50a362.  12.   V27  2;2  -  81  i/2. 


13.  4a;V(38^y.  16.   V64a4. 

14.  (9  a2_  27^2)1^    .  17^   8^25:^14^^-6. 

15.  (16a3-48a25  +  48«52_i(3  53)l.  18.  J^  a/64000  o^y. 
19.  Vi.  20.    V|.  21.    V|. 


60  COLLEGE   ALGEBRA 

The  student  will  here  notice  that  it  is  unnecessary  to  mul- 
tiply both  numerator  and  denominator  by  the  entire  denomi- 
nator, but  only  by  such  a  factor  as  will  make  the  denominator 
a  perfect  square. 

Thus    V 7  —  VI5  —  V-J-  •  14  =  V^  •  Vl4  =  1  Vl4 

44.    ^  V2-0  0- 

Here  the  denominator  must  be  a  perfect  cube,  and  the 
smallest  permissible  multiplier  is  therefore  5. 


TTpnpp     2  V  -2-1-  —  9  -^/JLOA^  —  9  V  -  J V  105 


=  2- J^V105  =  -1V105. 


23.    \—T7y — •  27.    \ 7^^- 


^  3  ^c^  0?  —  y^^  y^ 


—  X 


x-\-  y  l  +  a;^(l  +  xy 

97.    To  reduce  a  mixed  surd  to  an  entire  surd.     This  is 
another  application  of  the  same  principle  as  95. 

Thus,  2  ^  =  \/23 .  ^5  =  -v/iO. 

Also,  I V7  =  V| .  V7  =  VM. 

Reduce  the  following  to  entire  surds :  

1.  5V10.  3.  — 7  \— n:- 

a  —  0  ^ a-\-  0 


2«3/-2T2  ^       ^-^^    ^ 


2.   ^^<Ja%\  4 


J^^-f^ 


3  5  a;4-y  ^' 


IMAGINARY   NUMBERS  61 

98.  The  problem  of  reducing  a  number  of  surds  to  the 
same  order  may  be  illustrated  by  writing  them  with  frac- 
tional exponents,  and  then  reducing  these  exponents  to  a 
common  denominator.  Thus,  if  it  be  required  to  reduce  V2, 
V 5,  V7,  and  VTO  to  the  same  order,  we  have 

^/2  =  2^  =  2'^  =  '-</¥  =  ^■M; 
^5  =  53  =  5^  =  ^5^4  =  ^625; 
^7=7^  =  7r-=^r3  =  ^MB; 
-^10  =  10*  =  10^^  =  ^102  =  ^100. 

EXAMPLES 

Reduce  the  surds  in  each  of  the  following  groups  to  the 
same  order : 

1.  </5  and  -^7.  5.   ^(a  -  by  and  </{a  +  by. 

2.  VT  and  -v/TO.  6.   </2,  ^10,  and  ^5. 


3.  V^  and  </(^d.  7.   V|,  ^'f,  and  </^. 

4.  V2Wc  and  Vchl^.  8.   ^|,  -\/a%%  and  V2  a%. 

Note.  In  Example  8  the  student  will  see  that  though  Va-b^  is 
nominally  of  the  fourth  order,  it  is  equal  to  Vcib,  and  hence  should  be 
regarded  as  of  the  second  order.  Hence  the  three  may  be  reduced  to  the 
sixth  order,  instead  of  the  twelfth,  as  might  perhaps  be  supposed. 


9.  ^'l',  \§,  ^i  o\  and  ^5  d3. 

10.   \/— ■)  \'-,  \x^y^  and  v 


y      'X 

Definition.  Similar  surds  are  surds  which,  when  reduced 
to  their  simplest  form,  do  not  differ  at  all,  or  differ  only  in 
their  coefficients. 


62  COLLEGE   ALGEBRA 

ADDITION  AND   SUBTRACTION   OF  SURDS 

99.  For  reasons  which  will  become  apparent  in  the  con- 
sideration of  the  properties  of  quadratic  surds,  only  similar 
surds  can  be  united  into  one  surd  by  addition  or  subtraction. 
Therefore,  in  order  to  add  or  subtract  surds,  reduce  each  to 
its  simplest  form,  and  combine  the  coefficients  of  those  which 
are  similar. 

100.  EXAMPLES 

1.  Add  V6,  V2i,  Vf,  V15,  V|. 

Reducing  to  the  simplest  form  and  uniting  similar  surds, 
we  have 

V6+VM  +  V|  +  Vl5  +  V| 

=  V6  +  2V(5  +  lV6+Vl5  +  -iVT5 

=  (1  +  2  + 1)  V6  +  (1  +  -J)  Vl5  =  JgQ- V6  +  f  VT5. 

2.  Simplify 

VWM  4-  V50"^  _}_  2  aVJh  -7</¥-  3  Vl8^. 

</WM  4-  V50  a%^  4-  2  a</Sl)  -1</¥  -  3  Vl8^2 
=  SaVb-h5ab VWa  +  4.  a</b -1  h</b  -  9b  V2^ 
=  (3«  +  4«-75)\/6  +  (5a5-9  J)  V2^ 
=  (7  a  -  7  ^)-v/6  +  (5  a5  -  9  J)  V2^. 

Simplify : 

3.  V8  +  2VT8-3V32-14V2. 

4.  V20  +  V45-7V|. 


5.  3V2a  +  6V54a;-7aV16. 

6.  4v'32-5-v/162  +  14a/8T. 


IMAGINARY  NUMBERS  63 


11.  ,  V7  2;«  +  56  x*  +  112  a^  -  V3 13  -  18  x2  +  27  2;. 

12.  •</1876^V^  +  2a-!/v'48l?p-6^y-^|^ 

13.  ^2r^  +  ^24-A^|'  +  ^i. 


14.    4  a;  V68  x^y"-  +  5  «/ VlT  :r/  +  14  V60  xif. 


«/12  y 


16.     V2  2:2  +  4  2-?/  +  2  ?/   +  VI8  ?/2  _  12  :?;j/  +  2  x^ 

-V2a;2-8a;?/  +  8/. 

MULTIPLICATION  OF   SURDS 

101.  Two  methods  are  in  common  use  for  the  multiplica- 
tion of  one  surd  by  another.  They  are  exemplified  in  the 
following : 

1.  V2  X  V3  =  V2^<~3  =  V6,  (95) 

or  V2x  V3  =  2^x3^=6^  =  V6.  (90) 

In  this  example  both  methods  are  exceedingly  simple,  being 
nothing  but  direct  applications  of  the  elementary  principles. 

2.  V2x^2=^8xa/4=^32; 

or  V2  X  a/2=  2*  X  2^  =  2'-^*  =  2'^  =  -^  =  ^32. 

For  the  first  method  in  this  problem  it  is  necessary  to  re- 
duce the  surds  to  the  same  order  before  95  can  be  applied. 


64  COLLEGE   ALGEBRA 

The  second  method  is  a  direct  application  of  89,  since  the 
quantity   affected   by   the    exponent   is   the   same   in   both 
factors. 
3. 
■^6  X  -v/12=  ^1296  X  ^1728  =  a/1296  x  1728  =  ^2289488, 

or  a/6  X  v'12=  6*  X  12^  =  6^'^  x  12^'^=  (6*)^'^  x  (U^y'^ 

=  (1296y'^  X  (1728)T^ 
=  (2239488)!'^ 


=  V2239488. 

By  the  first  method  this  example  is  just  like  2 ;  but  the  sec- 
ond method  there  used  will  not  apply  in  exactly  the  same 
way,  since  the  quantities  affected  by  the  exponents  are  not 
the  same  in  the  two  factors.  Hence  it  is  necessary  so  to 
transform  the  factors  that  the  exponents  will  be  the  same,  in 
order  that  89  may  apply. 


4.  2V5x3V2  =  2V25x3V8  =  6V200, 


or 


2(5)^  X  3(2)^  =  6(5)^  x(2)*  =  6(25)*(8)*  =  6(200)*  =  6-^200. 

Since  the  product  of  a  series  of  quantities  is  the  same  in 
whatever  order  they  are  taken,  the  coefficients  are  multi- 
plied together  independently.  The  surds  are  then  multi- 
plied together  by  either  method. 

Simplify  : 

^      -^  5.    VlO  X  Vl5  X  V6. 

6.  a/12  X  V5  X  </S. 

7.  2VT2  X  3V15  X  4a/5. 


8.    3Vix4Vfx5V|. 


"  V-^# 


IMAGINARY  NUMBERS  65 


J2  \  ^2  \  ^2 


11.  Va^  —  52  X  V (a  +  5)2  X  va  —  ^. 

12.  Multiply  2  VS  -  5  V2  by  3  V3  +  4  V2. 

2  V3  -  5  V2 

3  V3  +  4  V2 
18-15V6 

+    8V6-40 
18-    7V6-40=-22-7V6. 

13.  Multiply  2  V3  -  5^2  by  3  V3  +  4  v^. 

2V3-5^ 
3V3  +  4^2 


18  -  15  Vl08 

18-    l</10S-20</4:. 

In  order  to  multiply  2V3  by  4v2  it  is  necessary  to  reduce 
them  to  the  same  order. 
Multiply : 

14.  3V3-4V2by  2V3-6V2. 

15.  3v'4-4^2by4^4-3^2. 

16.  2 Va5  —  2Vcd  by  2 Vac  —  2^bd. 

17.  3V^-2V^-4V«5by  2-v^-3V^. 

19.  2Vh  +  SVc  +  4:^d  by  V«  +  V5. 

20.  3  V^  -  4  V^2  by  v^  +  VP. 


66  COLLEGE   ALGEBRA 

DIVISION  OF   SURDS 

102.  The  methods  in  use  for  division  of  one  surd  by  an- 
other involve  the  same  principles  as  those  used  in  multiplica- 
tion, and  are  so  closely  related  to  these  methods  as  scarcely 
to  need  any  explanation. 

.—        q/ —       fi/ ■       fi/ ■        Q\a%^        elh      1  fi,— — 


s/—or      ^ab  _  '\/ah       -^aW-  _  h-\a^b  _  1  r,—^ 


r^-n         /~T       3/-nT       \  at)       W ao       -\  ab^      b-\  ayb      i  gz-tt- 
or       Va6  ^  V a^6  =  -^—^  =  ^-=:  X  -ir;=L  = t"  =  -  Va^b. 


In  the  first  method  the  surds  are  reduced  to  the  same  order, 
whence  division  is  readily  effected,  and  the  result  is  reduced 
to  its  simplest  form.  In  the  second  method  fractional 
indices  are  used,  which  are  reduced  to  a  common  denomi- 
nator, then  to  a  common  index,  whence  90  enables  the 
division  to  be  effected.  The  result  is  restored  to  its  surd 
form  and  simplified.  In  the  third  method  the  division 
is  first  indicated  by  writing  the  divisor  as  the  denomi- 
nator of  a  fraction.  Then  both  numerator  and  denomi- 
nator are  multiplied  by  that  quantity  which  makes  the 
denominator  rational.  This  does  not  change  the  value  of 
the  fraction.  The  multiplication  is  performed  as  in  95. 
This  last  method  is  known  as  the  method  of  rationalizing 
the  denominator. 


IMAGINARY  NUMBERS  67 


Simplify : 

2.  V24-f-  V6. 

3.  V24-VT6. 

4.  V50-V-V40. 

5.  V63-v'42. 

6.  V8^^12. 

7.  ^/24-V6. 

8.  'Vab  -^  "V^. 


EXAMPLES 

9. 

■yah -V-  Va6. 

10. 

V2  a6  -  -\/2  a26. 

11. 

3^^,2^^5-.y5^, 

12. 

2Va-5^3Va4-J. 

13. 

3^1 -V|. 

14. 

V6  4-Vl5-^2V3. 

15. 

VB  +  V5  -  2^/3. 

COMPARISON   OF   SURDS 


103.  The  relative  magnitude  of  a  series  of  surds  may  be 
readily  compared  if  tliey  are  reduced  to  the  same  order  ;  thus 
to  compare  Vo,  VlO,  and  V32,  we  reduce  them  to  the  twelfth 
order. 

V0=  ^15625;     ^10  =v  10000;      v'M  =  v'MTeS. 
A  comparison  of  these  results  shows  that 

-{/32  >  V5  >  ^yiO. 

INVOLUTION  AND   EVOLUTION   OF   SURDS 

104.  A  surd  may  be  raised  to  any  required  power  by  any 
of  the  methods  given  for  multiplication  of  surds.  Fre- 
quently it  is  advisable  to  express  the  surd  as  a  fractional 
power  of  the  expression  affected  by  the  radical,  and  then 
raise  it  to  the  required  power.     Thus 

Any  root  of  a  surd  may  be  obtained  by  a  similar  process. 
Thus  3,  ,  1   1  1       ,.,  


68  COLLEGE   ALGEBRA 


RATIONALIZATION 


105.  Definitions.  Rationalization  is  the  process  of  making 
a  surd  rational  hy  multiplying  it  hy  some  other  expression. 

The  expression  which  is  used  to  multiply  a  given  surd  in 
order  to  make  it  rational  is  called  a  rationalizing  factor. 

A  binomial  surd  is  a  binomial  in  which  one  or  both  of  the 
terms  is  a  surd. 

Two  binomial  quadratic  surds  are  called  conjugate  surds  if 
they  differ  only  in  the  sign  of  one  of  the  terms. 

Thus  4  -  V5,  x^  -i-yK  3  +  V7,  3  -  VT,  V^  +  V?,  and 
-\/y  —  Vsj  are  all  binomial  surds,  and  the  last  four  form  two 
pairs  of  conjugate  surds. 

Since  the  product  of  the  sum  and  difference  of  two  quan- 
tities is  equal  to  the  difference  of  their  squares,  it  follows 
that  the  product  of  two  conjugate  surds  is  always  rational. 

106.  The  method  of  reduction  of  a  fraction  whose  denomi- 
nator is  a  monomial  surd  to  an  equivalent  fraction  with  a 
rational  denominator  has  been  given  in  102. 

When  the  denominator  is  a  quadratic  binomial  surd,  the 
process  is  as  follows  : 

2  -  V3  ^  (2  -  V3)(3  -  V5)  ^  6  -  3 V3  -  2 V5  +  Vl5 

3  +  V5      (3  +  V5)(3-V5)  9-5 

6_3V3-2V5  + Vl5 

Also, 

■Va  -  3Vg  ^  (Va  -  3V^)(V^+  2V^)  ^a-  Vab  -6h 
Va-2V^~(Va-2V^)(Va  +  2VD~        a-4:b 

When  the  denominator  is  a  binomial  surd  of  the  nth  order, 
the  process  depends  upon  two  well-known  theorems  in  factor- 
ing, viz. : 


IMAGINARY  NUMBERS  69 

W7ien  n  is  odd.,  x^  ±  y'^  has  for  factors  x±y  and  x^~^  T  x^~'^y 

Whe7i  n  is  eve7i^  x^  —  y^  has  for  factors  x  ±y  and  x^~^  T  x^~'^y 
+  x''~^y^  T  •••  T  y""^     For  a  proof  see  86,  Ex.  10,  and  353. 

107.  EXAMPLES 

91 
1.    Rationalize  the  denominator  of 


3 --^4 


Therefore 


(3)3_  (^'4)3^  (3  _  ■^4)(32  +  3sy4  +  (V4)2. 
2 


2r3^  +  3V4  +  (:v4)^1         ^2[32  +  3V4  +  (V4)2-| 
"(3_^4)[32+3v/4  +  (^4)2]  33 -(^4)3 

2(9+  3a/^4  +  2v^2)  ^  18  +  6^4  +  4^/2 
"27-4  23 

2.    Reduce  — z =:  to  an  equivalent  fraction  with  a  ra- 

4/         ,       4/7  ->- 

tional  denominator. 

^  +V^a(A/6)2-(^5)3]. 


Therefore 


</a-\-</h 

(</'ay  -  (</^)^</b  +  </a(</by  -  (</hy 

(-^rt  +  ^6)  [(-y«)3  -  C</ay</b+  -\/^(a/^)2  -  (a/6)3] 

V«3  —  Va^  V5  +  Vrt  V62  —  VP 
(vay-(A/6)4 

->y^3  _  ^^  -I-  -^^2  _  ^p 

a  —  6 


70  COLLEGE   ALGEBRA 

3.    Express  — = ^-^  with  a  rational  denominator. 

Va;—  V«/ 

The  least  common  denominator  of  the  indices  2  and  3  is  6, 
and  therefore,  by  the  principle  above  stated, 

1  nereiore,  — r: ^^ 

Va;  — "V  ?/ 

(Vi)(^^)4  +  (^^/]    divided   by  (Vi- ^^)[(V^)5  + 
( V;r)4-v^  +  (  Vxf{</]/f  +  {-w'xfiVyf  +  Vx{</]/y  +  (^^)5] 
_  a{x^^\fx-\-o^^ y^x^x^y^Ar  xy  4-  ^xy\y  +  ?/  V^^) 


_  a(x^^x  +  :z^^  V  ;^  +  x^x^y^  -\-  xy  -\-  y^.x^y'^  -f-  y^y'^') 

x^  —  ?/2 

4.    Reduce  - — ^ z ^  to  a  fraction  with   a   rational 

V2-V8  +  V5 
denominator. 

Multiplying  the  numerator  and  denominator  by  V2  —  V8 

—  V5,  the  fraction  becomes 

2-(V3+V5)2^-6-2Vl5^3  +  Vl5 
(V2-V3)2_5  -2V6  V(3 

which  is  equal  to 

3  V6  4-  3 VTO  ^  V6  +  VTQ 

6  ~  2         * 


IMAGINARY  NUMBERS  71 

In  general  a  trinomial  expression  of  the  form  V5  +  Vy 
+  V2  can  be  rationalized  by  multi23lying  it  by  the  expressions 
■^x  —  V^  —  V2,  Vo;  +  V?/  —  V2,  Va;  —  V^  +  V2  ;  the  product 
of  the  first  two  is  a;  —  ( V«/  +  V^)^  =  x  —  y  —  z  —  ^'^yz^  and 
tlie  product  of  the  second  two  is  2;  —  ( V?/  —  V^)^  =  x  —  y  —  z 
+  2^yz^  and  the  product  of  these  partial  products  is  the 
rational  expression  (x  —  y  —  zy^  —  \  yz  =  x^  -\-  y"^  -\-  z^  —  2  xy 
—  1yz—2zx.  The  four  expressions  here  multiplied  together 
are  called  conjugate  trinomials.  It  is  to  be  noted  in  the  fore- 
going example  that  the  product  of  the  remaining  trinomials 
V2  +  V3  —  V5,  V2  +  V3  —  V5  is  2  V6,  which  shows  why  V6 
had  to  be  used  in  completing  the  rationalization.  In  general 
any  polynomial  quadratic  surd  may  be  rationalized  by  multi- 
plying it  by  the  product  of  all  of  its  conjugates. 

One  of  the  objects  of  rationalizing  the  denominator  of  a 
fraction  containing  surds  in  the  denominator  is  for  the  pur- 
poses of  calculation,  for  rationalizing  the  denominator  saves 

both  time  and  labor  and  conduces  to  accuracy. 

2 

5.    Thus  to  calculate  the  value  of we   have, 

V3+ V2 
after  rationalizing  the  denominator,  to  calculate  the  value  of 
2(V3-  V2),  which  is  equal  to  2(1.73205.-.  -1.41421...) 
=  2(.31784...)=  .63568...     The  student  may  compare  this 
with  dividing  2  by  1.73205...  +  1.41421...  or  3.14626... 
Rationalize  the  denominators  of  : 


6. 


V2-  a/3         g     V2+ V3 
V3  +  2  '    V5  -  VT 


,     6-V3                  a+V6 
7.    -'  9.    — ' =• 

12.  ^-^+^-^-Vi.      ,3. 


10.     ^'' 

-V6 

-  V5 

11     ^- 

_  V3+ V7 

V2 

H-  V3h-  VT 

Va  + 

V3- V2 

72  COLLEGE   ALGEBRA 

Calculate  to  four  places  of  decimals  the  values  of  : 

14.  . ^-  15.  2yi-V2 

V^-V2  2V8  +  V2 

16         ;^^     ,  given  that  V5  =  2.23607  .... 

V5-2 

PROPERTIES   OF  QUADRATIC   SURDS 

108.  Theorem.  A  surd  cannot  equal  the  sum  of  a  rational 
expressioii  aiid  a  surd. 

For,  if  possible,  let      Va  =  5  +  V^ 

where  5  is  a  rational  expression,  and  Va  and  V^  are  quad- 
ratic surds.     Squaring  both  members,  we  have 

r      a—lP'—c 
or  Vc= — , 

A  0 

that  is,  a  surd  is  equal  to  a  rational  expression,  which  is 
impossible. 

109.  Theorem.  If  a+V6  =  tf+V5,  where  a  and  h  are 
rational  numbers  and  V^  and  'Vd  are  surds,  then  a  =  c  and 
h=d. 

For,  by  transposition 

V^  —  c—  a-\-  V(/, 

and    by   the    preceding    theorem    <?  —  a  =  0,    and   therefore 
c=  a,  and  consequently  b  =  d. 

lia  Theorem.  If  ^ja-\-Vb=  Vx-\-  V^,  then  yja  -  V6 
=  Va:  —  V?/,  if  a,  5,  X,  and  y  are  rational,  V6  is  a  surd, 
and  a >  Vi  and  x>y. 


IMAGINARY   NUMBERS  73 

For,  by  squaring  we  obtain 

Therefore,  by  the  last  theorem, 

a  =  x+y, 

and  V^  =  2  ^xy. 

Subtracting  a  —  V3  =  x  —  2^xy  +  y. 

Extracting  the  square  root, 

V« — V5  =  ±  ( vi  —  v^), 

where  the  double  sign  is  determined  in  accordance  with  the 
conditions  of  the  problem. 

111.  To  extract  the  square  root  of  a  binomial  surd.  The 
method  of  doing  this  may  be  illustrated  by  the  following 
example : 

Find  the  square  root  of  16  +  2  V55. 

Like  every  other  number,  the  binomial  surd  has  two  square 
roots ;  but  since  the  given  binomial  has  both  terms  posi- 
tive, it  follows  that  the  positive  root  must  have  both  terms 
positive. 

Hence  we  may  let  \16  +  2  V55  =  Vo:  +  V^.  (1) 

Then,  by  110,  Vl6  -  2  V55  =  V^  -  V^.            (2) 

Multiplying  (1)  by  (2),    V256  -4-55  =  x-y, 

or  Q  =  x  —  y.                    (3) 

Squaring  (1),  16  -f  2  V55  =  x-\-  y  -\-^  V^, 

whence,  by  109,  16  =  x  -\-  y.                   (4) 

From  (3)  and  (4),  x  =  11,  and  y  =  5- 

Therefore  ^|l6^^^2VM  =  ±  ( VlT  +  Vo) . 


74  COLLEGE   ALGEBRA 

An  examination  of  the  above  process  shows  that  x  and  y 
are  the  two  numbers  whose  sum  is  the  rational  term  16,  and 
whose  product  is  bb^  and  that  a  similar  statement  must 
be  true  of  every  like  problem.  Hence,  when  the  numbers 
involved  are  not  too  cumbersome,  examples  like  the  above 
may  be  solved  by  inspection  as  follows  : 

Reduce  the  surd  term  to  the  form  in  which  its  coefficient 
shall  be  2.  Then  find  two  numbers  such  that  their  sum 
shall  be  equal  to  the  rational  term,  and  their  product  equal 
to  the  expression  under  the  radical  sign.  Extract  the  square 
root  of  each  of  the  numbers  so  found,  and  connect  the  results 
by  the  sign  of  the  surd  term. 

112.  EXAMPLES 

Extract  the  square  root  of  : 

1.  4  +  2V8.          5.    12-VT40.  9.  -V--^¥-- 

2.  4-2V3.          6.    11-V96.  10.  1  +  |V6. 

3.  5  +  2V6.         7.    57-12VT5.  ii.  3-|V5. 


4.    7  +  4V3.  8.    93-fV5400.  12.    2a-1^a^-hK 


-'4 


13.    m  +  '-      --/^^^ 


14.    11  a-Sb  +  W6a^-ab-b\ 

RADICAL   EQUATIONS 

113.  Definition.  An  equation  involving  one  or  more  irrational 
roots  of  an  unknown  nuinher  is  called  a  radical  equation. 

In  case  of  the  more  simple  types  of  radical  equations  the 
solution  is  readily  effected  by  the  ordinary  algebraic  processes. 
Yet  it  is  necessary  always  to  be  on  the  watch  lest  false  solu- 
tions be  obtained. 


IMAGINARY   NUMBERS  75 


1.  Solve  V^  +  7  =  8. 

Squaring  both  members,  rr  +  7  =  64. 

Hence  x  =  57. 

This  result  satisfies  the  original  equation,  and  therefore  is  a 
true  solution. 

2.  Solve  Va7  +  7  =  —  2. 

Squaring  both  members,  2;  +  7  =  4.  (1) 

Hence  x= -?,.  (2) 

According  to  the  laws  of  algebra  (1)  must  be  true  if  the 
original  equation  is  true,  and  (2)  gives  the  only  possible 
value  of  X  which  satisfies  (1).  Therefore  if  there  is  any 
true  solution  of  the  original  equation,  it  must  be  a:  =  —  3. 
But  trial  shows  that  this  value  does  not  satisfy  the  original 
equation.  Hence  we  are  forced  to  conclude  that  there  is  no 
true  solution  of  that  equation. 

In  fact  (1)  would  likewise  have  been  obtained  if  we  had 
started  with  the  equation  Va;  +  7  =  2,  and  we  have  really 
obtained  the  solution  of  this  latter  equation  instead  of  the 
one  which  was  given. 

Some  writers  avoid  this  difficulty  by  regarding  all  radical 
signs  as  affected  by  the  double  sign,  as  ±  ^x-\-  7.  But  this 
merely  transfers  the  difficulty  to  the  later  applications  of 
these  principles. 


3.    Solve  V2  x-{-l  —  ^x  —  b  =  0. 


Transposing,  V2  x-\-l  =  V2;  —  5. 

Squaring,  2  x-\-'i  =  x  —  5. 

Hence  x=  — 12, 

which  satisfies  the  original  equation,  and  is  therefore  a  true 
solution. 


76  COLLEGE   ALGEBRA 

4.  Solve  Va;  +  7  +  -Vx-5-  10  =  0. 

Transposing  so  as  to  have  one  radical  alone  in  one  member, 

VxT^  =  10  -  -Vx -  5. 
Squaring,  a;  +  7  =  100  +  a;  -  5  -  20Vx-5. 

Transposing  and  combining, 

20  V^^^  =  88, 
or  5-Vx-b  =  22. 

Squaring,  25(x  —  5)  =  484. 

Reducing,  x  =  242^^. 

This  satisfies  the  original  equation,  and  is  a  true  solution. 

5.  Solve  V^  4-  7  +  V:r  —  5  +  Va;  —  2  —  Va;  +  6  =  0. 

It  is  not  practicable  to  separate  one  radical  in  this  problem, 
since  that  would  leave  three  in  the  other  member.  Hence  it 
is  better  to  leave  two  in  each  member. 

Thus,  Vx+  7  +  V:r-5  =  V^+IT-  Vx-2.  (1) 

Squaring,  x+7  -\-  x—  5  +  2Vx^-{-2x  —  Sb 

=  x+6  +  x-2-2^x^-h-itx-12.       (2) 

Transposing  and  reducing, 

-Vx^+2x~S5  =  l--Vx'^-\-4:x-12.  (3) 

Squaring   again, 

af^^2x-S5  =  l  +  x^+4:x-12-  2 V^Tl^^H^.     (4) 

Transposing  and  reducing, 

Vx^-{-4:x-12  =  x-\-12.  (5) 

Squaring  again,     x^ -i- 4:  x  —  12  =  x^ -{- 24:  x -{- 144.  (6) 

Hence  a:  =—7.8.  (7) 


IMAGINARY   NUMBERS  77 

This  is  the  only  solution  obtainable,  and  yet  it  does  not 
satisfy  the  original  equation.  Hence  that  equation  has  no 
true  solution. 

In  fact,  by  substitution  x=  —  7.8  will  be  found  to  satisfy 
equations    (6),   (5),   and    (4),    but    instead    of    satisfying 

(3)  it    will   be    found   to    satisfy    the    companion   equation 
—  Vrc^ -\-2x—  35  =  1  —  ^x^ -\-AlX—  12,  from  which  equation 

(4)  could  equally  well  have  been  derived. 

Further  it  is  observed  that  a;  =  —  7.8  satisfies  the  equation 


V^~+T  +  Va:  —  5  =  Va;  +  (3  H-  ^x  —  2, 

which  on  squaring  would  make  the  sign  of  the  radical  in  the 
second  member  of  (3)  positive,  and  when  this  is  squared  it 
will  lead  to  (4). 

Remark.  In  general,  it  is  seen  that  by  squaring,  results  are  obtained 
which  belong  to  equations  like  the  original  except  that  the  signs  of  some 
of  the  terms  are  changed.  In  fact,  squaring  the  equations  x  =  a  and 
X  =  —  a  leads  to  the  same  equation,  x^  =  a?-.  Roots  which  are  introduced 
by  squaring  are  sometimes  spoken  of  as  "  extraneous  roots,"  but  they  are 
really  no  roots  at  all  of  the  original  equation,  and  whether  the  solution 
obtained  is  an  actual  root  or  not  can  only  be  determined  by  substitution. 
The  work  should  never  be  considered  as  finished  until  the  character  of 
the  solution  has  been  ascertained. 

114.  EXAMPLES 

Solve  the  following  equations : 

1.  V3  a;  -  7  =  25. 

2.  Vo-  2j;-3V2-  32:  =  0. 

3.  V2  2;  -  14  +  V^'  =  V2  2'  +  7. 

4.  V2  :r  -  14  -  V^  =  V2.f  +  7. 
6.  Vo  a;  -  1  +  V2  2;  +  5  =  12. 


78 


COLLEGE  ALGEBRA 


6.  Va:2  -  32^  +  10  -  V:r^  +  2  a:  +  12  =  2. 


7.  Vb-10x-Vl-4:x=^5x-hU. 


9.  V:?;  +  5  +  V2  2:  +  8  =  0. 


10.  Va;  +  5  +  V2  :z:2  _^  7  2;  -  15  =  0. 

11.  ^2  +  2: Va;2  +8a;  +  40  =  2  +  a;. 

12.  2:  + Va;2  + V5^2T^3^Tl  =  l. 

13.  x-\--\lx^-^dV5x^-dx  +  '3  =  S. 
6 


14. 


15. 


V3^  +  5   V3^  +  1 

V3^-l~  V8^-2' 


5. 


Suggestion.     Regard  as  a  proportion  and  apply  addition  and  sub- 
traction, by  42. 


16. 


^jV6x-h2        VV5a;-19 
■\-V'bx—lc>      yV5x  +  13 


V3  x  —  V5  X  -j- 1 


18. 


19. 


VI2  x-\-^Sx^-\-5x-{-lS      ^^Q 
Vl2^-  V8a;''^  +  5:r+13 


V2  a;  +  5  +  V3  a;  +  10      V2  a;  4-~8  +  V^V-h  26 
V2  a;  +  5  -  V3a;  +  10  ~  V2a:H-8  -  V5  a:  +  26 


IMAGINARY  NUMBERS  79 

COMPLEX  NUMBERS 


115.  In  87  numbers  like  V—  3,  V—  5,  etc.,  have  been  in- 
troduced. Since  V— 3  =  V3 V— 1,  V— 5=VoV— 1,  and 
in  general  V—  ^  =  VaV—  1,  it  is  seen  that  all  such  numbers 
can  be  expressed  in  terms  of  V—  1.  The  symbol  i  will  be 
used  to  denote  V— 1;  thus  every  pure  imaginary  number 
can  be  expressed  in  the  form  of  5^,  where  5  is  a  real  number 
and  i  the  unit  of  pure  imaginary  numbers. 

116.  From  the  definition  of  i  and  from  the  theory  of  ex- 
ponents we  have  for  the  powers  of  i  the  following : 


zi  = 

h 

v^  = 

-1, 

^3  = 

-^^ 

i'  = 

1, 

2*"  = 

1, 

{in+1  ^ 

h 

^•4n+2  _ 

-1, 

^•4h4  3  _ 

—  ^. 

Thus  the  value  of  every  integral  power  of  i  is  contained 
among   +  ^,  —  z,  +1,  —  1. 

117.  In  the  case  of  a  complex  number  a  +  hi^  if  a  =  0  and 
h=^0  the  complex  number  becomes  a  pure  imaginary  number  ; 
if  6  =  0  and  a  ^  0  it  becomes  a  real  number,  and  ii  a  =  b  =  0 
it  is  equal  to  zero.  If  either  a  or  h  or  both  become  infinite, 
the  number  becomes  infinite.  It  is  seen  that  the  complex 
number  is  a  twofold  generalization  of  the  elementary  number 
concept. 

In  making  the  extension  consistency  requires  that  the 
extended  number  be  subject  to  the  fundamental  laws  of 
algebra. 


80  COLLEGE   ALGEBRA 

118.  Definition.  Two  complex  numbers,  a  +  hi  and  a  —  hi, 
differing  only  in  the  sign  of  the  pure  imaginary  part,  are 
called  conjugate  complex  numbers.  It  is  seen  that  the  con- 
jugate of  any  complex  number  is  obtained  by  changing  i 
into  —  i ;  thus  a  +  hi  is  the  conjugate  of  a  —  hi  and  a  —  hi  is 
the  conjugate  of  a+  hi.     So  also  are  hi  and  —  hi  conjugates. 


119.  Definition.  The  expression  V^^  -\-  ^2  taken  with  the 
positive  sign  is  called  the  modulus  of  a  +  hi,  and  the  abbrevi- 
ation mod  will  be  used  to  denote  "the  modulus  of."  Thus 
mod  (a  +  iz)  =  Va^  +  ^^.  *  If  5  =  0,  that  is,  if  the  complex 
number  is  wholly  real  the  modulus  reduces  to  Va^  which  is 
simply  the  absolute  or  numerical  value  of  a. 

It  is  seen  that  mod  (^a  +  hi)  =  mod  (a  —  hi'),  since  each  is 
equal  to  Va^  +  h^. 

120.  Theorems.  The  sum,  difference,  product,  and  quotient 
of  two  complex  numhers  is  in  general  a  complex  number.  For 
let  a-^  +  h^i  and  a^  +  h^i  be  two  complex  numbers,  then  their 
sum,  which  is  a-^-\-  a^-\-  (h^  +  h^i  is  in  general  a  complex 
number.  The  difference,  which  is  a-^^  —  a^-\-  (h^  —  h^i,  is  like- 
wise a  complex  number.     The  product  is 

(a-^a^  —  h^^  +  (^a^^  +  ^^i)** 

and  is  also  a  complex  number.     The  quotient 

^1  +  ^1^*  _  ^^1  +  ^iO(^2  ~"  ^2**)  _  ^\^2  +  ^1^2  I  (^2^1  ~  '^1^2)^* 
«2  +  K^  ~~  (^2  +  h^)  C«2  -  ^20  ~    <^2  +  K  ^i  +  ^2 

is  in  general  a  complex  number. 

From  this  it  follows  that  any  rational  integral  function  of 
a  complex  variable  is  a  complex  number ;  thus  /  (a  -}-  hi)  is 
of  the  form  A  4-  Bi,  where  A  and  B  are  real. 

*  The  modulus  of  a  number  is  also  denoted  by  placing  the  number  between 
two  vertical  strokes.     Thus  the  modulus  of  «  +  hi  is  denoted  by  |a  +  hi\. 


IMAGINARY  NUMBERS  81 

It  is  also  easily  seen  that  if 

then  f  (a  —  hi^  =  A  —  B% 

and  therefore, 

mod/  (a  -f  hi)  =  mod/  (a  —  hi)  =  -ylfi^a  +  hi)  f(^a  —  hi). 

SOME   PROPERTIES   OF  CONJUGATE   COMPLEX  NUMBERS 

121.  The  sum  of  tivo  conjugate  complex  numbers  is  real  and 
equal  to  twice  the  sum  of  the  real  part  of  either. 

Thus  (a  +  hi)  -\- (a  — hi)  =2  a. 

122.  The  difference  of  two  conjugate  complex  numhers  is  a 
pure  imaginary  and  is  equal  to  twice  the  imaginary  part  of  the 
one  which  is  used  as  minuend. 

Thus  (<^  +  hi)  —  (a  —  hi)  =  2  hi. 

123.  The  product  of  two  conjugate  complex  numhers  is  real 
and  positive  and  is  equal  to  the  square  of  their  modulus. 

Thus  (a  +  hi)  (a  —  hi)  =  a^  +  V^. 

The  conjugate  of  f(a-\-hi)  is  f(a  —  hi)^  as  is  apparent 
from  118. 

IDENTITY  THEOREMS   FOR    COMPLEX   NUMBERS 

124.  If  a  +  hi-=  0,  then   a  =  0  and  h  =  0 ;  for  if   not,  we 

would  have  , . 

a  =  —  6^, 

that  is,  a  real  number  equal  to  a  pure  imaginary  number, 
which  is  impossible  ;  therefore  a  =  5  =  0,  that  is  if  a  complex 
numher  vanishes^  hoth  the  real  part  and  the  pure  imaginary 
part  must  vanish  together. 


82  COLLEGE   ALGEBRA 

If  a-^  +  h-^i  =  a^  +  b^i,  then  a-^  =  a^  and  h-j^  =  h^; 

for  ^j  —  «2  +  (^1  ~  ^2)*'  ~  ^' 
therefore,  by  124,  a^  —  ^2  =  ^i  —  ^2  =  ^' 
or  a-^  =  a^     and  ^-^  =  ^g, 

that  is,  {f  tivo  complex  numbers  are  equals  the  real  fart  of  the 
one  must  equal  the  real  part  of  the  other  and  the  imaginary  part 
of  the  one  equal  the  imaginary  part  of  the  other. 

125.  EXAMPLES 

1.  Find  the  sum  of  2  +  3  i,  V2  +  V5  ^,  VS  —  V2  2,  1  —  2  i. 

2.  Multiply  5  -  2  z  by  3  +  4 «. 

2  4-  5z 

3.  Find  the  value  of  ^—^ 

Q  —  I 

4.  Find  the  value  of 


V2  +  V3z 

5.  Find  the  value  of  ( V2  —  V3  iy. 

6.  Find  the  value  of x 


3  +  2  1  +  32 

7.  Find  the  value  of  3  a;^  —  2  2  +  2  when  2  =  2  +  2. 

8.  Find  the  value  of — — when  z  =  1  -\-2i. 

GRAPHIC   REPRESENTATION   OF   COMPLEX  NUMBERS 

126.  In  order  to  develop  thoroughly  the  theory  under- 
lying the  graphical  representation  of  a  complex  number,  it 
is  necessary  to  establish  a  number  of  fundamental  principles. 
For  the  sake  of  those  who  do  not  care  to  take  up  the  details 


IMAGINARY  NUMBERS 


83 


of   these  principles   the  following   customary   treatment   is 
given : 

If  the  line  OM  is  assumed  to  have  the  length  unity,  the 
student  is  already  familiar  with  the  method  of  representing 
+  5  by  the  point  P,  and  —  5  by  the  point  P\  which  is  at 
the  same  distance  from  0  in  the  opposite  direction.  But 
—  5  =  -}-5(— 1),  and  OP^  may  be  regarded  as  obtained  by 


Fig.  13. 


revolving  OP  about  0  through  an  angle  of  180°  in  the 
direction  indicated  by  the  arrow  head.  That  is,  the  factor 
"  —  1 "  may  be  regarded  as  turning  the  line  OP  through 
this  angle  of  180°. 

In  order  to  obtain  a  consistent  method  of  representing  the 
quantity  5V— 1,  we  note  that  —  5  =  5  (V^^)^  or  5  z^. 
That  is,  if  +  5  is  multiplied  by  i  twice  the  result  is  —  5. 
If  the  multiplication  by  i  twice  revolves  the  line  through 
180°,  consistency  demands  that  each  multiplication  by  i  must 
revolve  it  through  90°,  so  that  5  V—  1,  or  5  z,  is  represented 


84  COLLEGE   ALGEBRA 


hj  P" .  Likewise,  the  representation  of  —  5V— 1,  or  5  z^, 
requires  that  the  line  be  revolved  through  an  angle  of  three 
times  90°,  or  270°.  Hence  —  5  V—  1,  or  —  5  i,  is  represented 
by  P'". 

Since  the  ordinary  complex  number  is  the  sum  or  differ- 
ence of  a  real  number  and  a  pure  imaginary,  the  method  of 
its  representation  must  be  analogous  to  that  of  the  sum  or 
difference  of  two  real  numbers.  Thus,  to  obtain  that  of 
5  +  2,  two  units  are  measured  to  the  right  of  P,  giving  Q ; 
and  that  of  5  —  2  is  two  units  to  the  left  of  P,  or  Q'.  So 
5  +  2  ^  is  two  units  above  P,  or  i^,  and  5  —  2  ^  is  two  units 
below  P,  or  R' .  The  same  results  would  be  obtained  by 
first  laying  off  the  imaginary  units  from  0  upward  or  down- 
ward, and  then  measuring  five  units  to  the  right. 

Addition  or  subtraction  of  complex  numbers  is  accomplished 
by  a  similar  process.  Thus,  to  obtain  5+2  ^  +  (—  3  +  4  2), 
three  units  are  measured  to  the  left  of  M  and  four  are  then 
measured  upward,  giving  S  as  the  representation  of  the 
sum.  The  same  point  aS'  is  obtained  by  taking  the  point  T^ 
which  represents  the  number  —  3  +  4  z,  and  finding  the  fourth 
vertex  of  the  parallelogram  of  which  T^  0,  and  H  are  the 
other  three. 

Since  subtraction  consists  merely  of  the  addition  of  the 
corresponding  negative  number,  it  does  not  require  a  separate 
discussion. 

It  is  to  be  observed  that  in  the  above  a  complex  number  is 
represented  by  a  point  whose  abscissa  is  the  real  portion  of 
the  complex  number  and  whose  ordinate  is  the  coefficient 
of  i  in  the  imaginary  portion.  That  is,  the  complex  number 
is  regarded  as  written  in  the  form  x  +  ii/. 

By  many  authors  it  is  considered  a  sufficient  justification 
of  the  whole  method  of  representation  merely  to  state  that 
since  every  point  is  represented  by  two  numbers  (coordinates) 


IMAGINARY  NUMBERS  85 

and  every  complex  number  x  +  iy  involves  two  independent 
real  numbers,  every  complex  number  may  be  represented  by 
a  perfectly  definite  point  in  the  plane,  and  conversely  every 
point  in  the  plane  represents  a  perfectly  definite  number. 
Real  numbers  are  given  by  points  on  the  a:-axis,  and  pure 
imaginaries  by  points  on  the  j/-axis. 

Note.  Instead  of  regarding  5  +  2  i,  —3  +  4  {,  and  their  sum  as 
represented  by  R,  T,  and  S,  respectively,  many  authors  regard  them  as 
re2:)resented  by  the  lines  OR.,  OT,  and  O-S",  respectively.  In  this  form 
the  student  of  physics  will  recognize  the  parallelogram  of  forces  and  the 
law  of  the  combination  of  motion.  The  lengths  of  these  lines  are  then 
said  to  represent  the  moduli  of  these  numbers,  and  the  angles  POR, 
POT  J  and  POS  represent  their  amplitudes. 

For  the  sake  of  those  who  desire  a  more  scientific  treat- 
ment of  these  topics  the  following  discussion  is  given,  based 
upon  more  fundamental  principles. 

127.    If  we  take  a  point  0  in  a  straight  line  and  lay  off 

OP 
from  0  a  positive  unit  OM.  then  the  ratio    of  the  sesr- 

^  OM  "" 

ment  OP,  determined  by  a  third  point  P  in  the  line,  to 

the  segment   OM  is  some   real  number,  positive   or  nega-. 

tive  according  as  P  and  31  are 

on  the  same  or  opposite  sides 

of  0.  ^^x 

In    this   case    OP   and   031 
coincide  in  the  same  geometric  ^^^  ^ 

line,    though   they    may    have 
opposite  directions.    Moreover, 

it  is  seen  that  to  every  point  P  in  the  line  corresponds  one 
and  only  one  real  number,  and  conversely  to  every  real 
number  corresponds  one  and  only  real  point  in  the  line. 
Thus  all  real  numbers,  and  only  real  numbers,  have  their 
representation  in  the  line. 


86 


COLLEGE   ALGEBRA 


Now  let  us  consider  the  interpretation  of 


OP 
OM 


where  OP 


and    OM  do  not  lie  in  the  same  geometric  line  and  where 
account  is  taken  of  the  angle  from  OM  to  OP  as  well  as  of 

the  lengths  of  the  lines.  This  is 
evidently  a  generalization  of  the  pre- 
ceding case,  and  it  is  apparent  that 
the  ratio  cannot  be  real  unless  OP 
and  OM  are  made  to  coincide  again. 
We  have  here  an  extension  of 
number,  and  in  defining  the  laws 
of  operation  which  it  shall  obey,  we 
are  only  restricted  to  consistency 
with  previous  definitions,  such  that  when  this  number 
becomes  real  no  contradictions  with  previous  definitions  exist. 

Definition.  We  define  the  number  as  determined  when  the 
ratio  of  the  lengths  OP  and  OM  is  determined  and  when  the 
angle  MOP  is  determined. 

OP  0'  P' 

Thus  ----  and  —— — —  define 


Fig.  15. 


OM    '     O'M' 

the  same  number  if  ratio  of 

the  length  of  OP  to  OM  is 
equal  to  the  ratio  of  the 
length  of  O'P'  to  O'M',  and 
the  angle  MOP  is  equal  to 
the  angle  M'O'P'. 

We  adopt  geometric  addition, 

that  is  OE-\-EQ=OQ, 

OQ  ^  OR 
OM 


Fig.  16. 


and 


RQ 


OM^  OM' 


and  also 


OR      OM^OR 
OM^  OQ      OQ' 


IMAGINARY   NUMBERS 

as  a  definition  of  multiplication.     These  are 

evidently  extensions  of  the  same  operations 

when  OR  and  OQ  coincide,  and  all 

of  them  are  consistent  with  previous 

definitions  when  the  number 

considered     becomes     real, 

that  is  when  OR  and  OQ  lie 

in  the  same  line.  ^^^'  ^'' 

128.    Let  the  line  chosen  in  127  be  the  axis  of  x. 
Then    OP 


R 


OM 


=  X.     Erect  a  perpendicular  OQ  to  OX  2ii  0,  such 


that  the  ratio  of  the  lengths  -p~  =  y.     Take  S  such  that  the 

.  OS 

ratio  of  the  leng'ths  —  is  equal 

^        OQ        ^ 
to   the   lenecth    ratio    — ^  =  y. 

Then  the  number  represented  by 

OS 
M         -TT^  is  equal  to  that  represented 

-*^X  OQ     g^ 

by  y—^  angles  as  well  as  ratios 

of  lengths  being  taken  into 
account,  since  both  have  the 
same  length  ratio  and  both  have 
a  positive  right  angle  for  angle. 

OS 


;Sh 


T 

Q 

o 


V 


Fig.  18. 


Hence 


and 


OS 


OQ 
OM 

[0Q\' 
\OMJ 


OQ' 

OS 

OQ 


OQ  ^  OS 
OM      031 


But  -~^  is  real  and  negative  and  equal  to  —  y^,  since  by  con- 


struction the   length    0^  is   a  mean   proportional   between 
that  of  O/S' and  Oitf.     Hence      /  QQY 


88 


COLLEGE   ALGEBRA 


Therefore 

Similarly, 
the  length  OQ. 


OQ  ^     . 
OM     ^'" 


OM 


=  —  yi  if  the  length  OQ'  is  equal  to 


Thus   it  is   seen   that   all   pure  imaginary 
numbers  are  represented  by  points  in  the  axis  of  y. 
In  general,  since 

OP  =  OM'  +  M'P, 


and 


OP  ^  OM'     M'P 
OM      OM       OM 


we  have 


OP 
OM 


OM'      OP' 
OM       OM' 


X  +  yi. 


where  x  is  the  abscissa  of  P  and  y  is  its  ordinate. 

129.    Denote  the  number 


have 


qp_ 

OM 


by  z.      We  have  seen 
OP 


(128)    that    ^  =  X  +  yi. 

Therefore  z  =  x  -\-  yi.     Let 
OP  =  r.        The     ratio     of 


lengths 


OP 
OM 


=  r,    and   also 


r  =  -Va^  4-  y^.  The  angle 
MOP  =  cf>  is  called  the  am- 
plitude or  argument  of  the 


X 


Fig.  19. 

z=  x  +  yi=  r  (cos  <f>-\-  i  sin (^) . 


complex  number.     Snice  - 

T 

^  y      . 
=  cos  (^,  and   -  =  sm  <^,  we 


IMAGINARY   NUMBERS 


89 


Geometrical  interpretation  of  the  four  fundamental  opera- 
tions upon  complex  numbers  can  be  given.  Any  two 
complex  numbers  may  be  constructed  with  a  common  de- 
nominator equal  to  the  positive  linear  unit  OM. 

130.  The  addition  of  two  complex  numbers  has  already  been 
treated  in  the  definition  127. 


Thus  if 


and  "  -  ^^2 

2—  -Q]^' 


^1  +  ^2-  031'^  OM 


0I\     PJ\_OP^ 
OM'^  OM  ~  OM 


=  z 


3^ 


where  P^P^  is  drawn  through  Pj 
parallel  to  OPg-  Thus  the  numer- 
ator of  the  sum  is  formed  by  com 
pleting  the  parallelogram  on  OP^ 
and  OP 2  as  sides  and  taking  the 
diagonal  from  0. 
Since 

OP2  +  OP^  =  OP  2  +  P^P^  =  OP3, 


OP 


.^OP, 


OP. 


_    ^^   2 


+ 


A^3 


OP. 


OM  '   OM      OM  '    OM       OM 


or 


^2  +  ^1  =  2 


3' 


therefore 


2i  ~r~  Ziy  —  Ze)  ~r"  ^ 


1' 


and  the  addition  of  complex  numbers  is  commutative. 

The  difference  of  tivo  complex  numbers  z^ 
structed  geometrically.     Let  z.^  =  x^  +  y^^  and  z^  =  x^-\-  y^i. 


2j  can  be  con- 


90  COLLEGE   ALGEBRA 

OM      OM' 


^2  ~  ^1  — 


Therefore 


~  OM^  OM      OM' 

OPi  ,  PiP.  _  OP. 
OM^  0M~  OM' 

qp^_qp^_pj\ 

OM      OM     OM  ' 


Thus   the    numerator   of   the  difference   is  constructed   by 
drawing  a  line  from  P^  to  Pg* 

131.    The  product  of  two    complex   numbers  can   be    con- 
structed as  follows : 

Let  .,  =  -^  and  .,  =  ^  =  ^  •  (Fig.  22) 

jj.  length  OP3       length  OPg  mod  (^2^1)      mod  z^ 

^^^^     length  OPi  ^  length  OM'  ^^      mod  z^     ^       T^' 

mod  (^2^1)  =  mod  z^  mod  Zp 
whence  mod  (z-^z^)  =  mod  z^  mod  z^. 

Therefore  mod  (z2Zi)  =  mod  (^z^z^~). 

By  construction,  amplitude  (^2^1)  =  amplitude 
2|  +  amplitude  z^,  therefore  amplitude  (^z-^z^^ 
=  amplitude  z^  +  amplitude  z^ 

Therefore    amplitude   CH^-^^  =  ampli- 
tude (2^122)- 

Therefore  z^z,^  =  z^z^  and  the 
multiplication  is   commutative,    ^ 
though  a  different  geometrical  Fig.  22. 

construction  would  be  made  for  z-^z^. 


IMAGINARY  NUMBERS 


91 


OP2' 


^2  ~  ^3^1- 

mod  z^  =  mod  z^  mod  2^, 


Thus  the  product  is  formed  by  constructing  the  sum  of 
the  amplitudes,  and  finding  a  line  equal  in  length  to  the 
product  of  the  lengths  of   OP^  and  OP2'  Xhws, 

0P^_   r 

(Fig.  23) 
132.    In  the  case  of  the  quotient  of  two 
complex  numbers^ 
let 

therefore 

Then 
and 


'     or^i  =  ^,orr3=0Pi.0P2. 


20  = 


Fig.  23. 


(131) 


amplitude  ^2  =  amplitude  z^  +  amplitude  z-^. 


Therefore     mod  23  =  2 


mod 


and 


amplitude  z^  =  amplitude  z^^ 
—  amplitude  z^. 

Thus  the  amplitude  of  the  dividend 
minus  the  amplitude  of  the  divisor  is 
to  be  constructed,  and  a  line  r  is  to 
be  obtained  whose  length  r^  satisfies 
the  equation 
OP 


2  _  ^3 


Fig.  24. 


OM 


=  r. 


3* 


(Fig.  24) 


133. 


OP, 

EXAMPLES 

Simplify  the  following  expressions  and  write  the  results, 
(1)  with  positive  exponents ;  (2)  in  integral  form ;  (3)  in 
simplest  form  with  radicals  : 


1. 


-2  ,  3 


-  5   3 

r-3/,4\¥ 


7-^  •»2 


2. 


i      1  .   _2  .  „     _..    _„ 

tn^n   \  ^    fm  ^n  ^^  ^ 


-h    ^ 


2  * 
p^q- 


92  COLLEGE   ALGEBRA 

3.    x^x^  where  a  and  h   are  the  roots  of  the  quadratic 
equation  y^  —  3  «/  +  2  =  0. 

he 

X{a-c){a-h) 

5. 


1                    1 

4. 

X{a-b){a-c)    .  ^(6-c)(6-a) 

1 

X{c-a){b-c) 
a                           b 

6. 

X{a-b)(a-c)   .  x(b-c)(b-a) 

c 

ab 


Xib-cKa-b)  .  x^<^-('Kb-c) 


X  (c-a)(b-c) 
1    1 

7.  x^x^  where  a  and  b  are    the   roots   of   the    quadratic 
equation  y^  —  4  y  +  4  =  0. 

8.  a^  -7-x^   where  a  and  h  are  the  roots  of  the  quadratic 
equation  ^^  _  ^  _j_  2  =  0. 

8^(128)^  +  V24  .  VT5  •  V2(3 


8-^nys^' 


</16^</2-</'6:l-</S2 


,,      V2  +  V3  C"^"^^)^- 

10.      — —  .  11^  „i        m-ni- 

2V 2  +  V 3  Xm-n  '        Va 


IX-y   Vj-^r    ,-\V-^Z         y~Z\y^x/ 


,2+X 


13.  (V2  +  l)^.  15.    (v'2-2V3)3. 

14.  (3-' -  2^ -}- 3^)2.  16.    aJi9  +  8V3. 

17.    [15-4(14)2]i 

19.  ^V3  + V5-2\/15. 

20.  [(48)^+(162)^  +  12(6)^]i 


IMAGINARY  NUMBERS  98 

Rationalize  the  denominators  of  the  following  fractions : 
21.  •  -t-  V- 


22. 


23. 


24. 


25. 


V8 

+  2V2 
5-3V2 

V2 

-  V3  + 

V5 

x  + 

.2V^ 

^x 

-v^ 

Va 

+  V5- 

V? 

Va 

-V3^  + 

V^ 

yJx 

+  2V^- 

-3V^ 

26 

3  + V-2 

V5-V-8 

27. 

7_5V-8 
3+2V-5 

28. 

11 +z 
7  —  5  ^  * 

29 

2 

V2  —  V3  i  —  V5  i 

QH 

^\fa  +  V^i  —  Vca 

2  V^  +  3  V^  —  -^x  Vai  —  'Vbi  —  Vc 


Find  the  numerical  value  to  4  decimal  places  of  the 
following  expressions,  given  V2  =  1.41421,  Vo  =  1.73205, 
V5=  2.23607,  V(j  =  2.44949,  V7  =  2.64575,  and 
V30  =  5.47723. 

31.     J   ,.  33.    2-'^^^ 


2V7  4-V6  2V2-3V3 

32.  _2±V3_^  3,_       5 


2V3-3V2  V2-V3  +  V5 

V2  -  V3  +  V6 


35. 


V2  +  V3  +  V6 


Find  the  value  of  x  in  the  following  equations : 

36.    V^  +  V2;  —  3  +  Vo;  +  5  =  0. 


37.    2V^^^+ V4a;-3- VlOa;+ll  =  0. 


94  COLLEGE   ALGEBRA 


38.  +  V8a:=  V3a;-2. 


39. 


(jx  +  5)^  -(x-  ly  ^  (^  -  2)^  -  {x  -  3)^ 


40.    f^-=^Y+(:^^+3)^  = 1-^. 


41. 


42. 


(flg  +  a:)^      _       (a  —  xy 
a*  +  (a  +  a;)^      a^  -  (a  -  o;)^ 

(2;  +  2  a)*  +  (a: -2  a)*      ^a 


43.  — ==— ' ^^=1:  H ==^ =  4Va;^  —  1. 

Va^  +  1  -  Va;2  -  1      Vx'-^  +  1  +  Va;2  _  1 

44.  (a^  +  x~^)^  =  (a^  +  a;)i 


45.  Vl3  +  :i:"-^+ V2.r2  +  12=  V2;2  +  29. 

46.  VS^HhTOI  —  V2^^^  =  V2:  —  z.  • 


47 


.    Va:  +  4  z  —  V:r  =  XJ  2  a: — . 


2 

48.  Va^  —  y^x  -\-  VI  —  a;  =  z. 

Find  the  square  root  of : 

49.  1-2V^T.  51.    4-6zV5. 


60.    12^V6-6.  52.    l-2zVl32. 


CHAPTER    VII 

PROGRESSIONS 

ARITHMETICAL   PROGRESSION 

134.  Definitions.     An  arithmetical  progression  (A.  P.)  is  a 

8uccessio7i  of  teryns  each  of  ivhlch  is  formed  from  the  preceding 
hy  the  addition  (^algebraically^  of  the  same  constant. 

This  constant  is  called  the  common  difference  of  the  terms. 

It  is  obvious  that  each  term  may  be  formed  from  the  fol- 
lowing term  by  taking  away  the  common  difference. 

The  entire  number  of  terms  in  an  arithmetical  progression 
is  unlimited.  In  what  follows  only  a  finite  number  of  con- 
secutive terms  of  the  A.  P.  will  be  considered. 

Let  a  denote  the  first  term,  d  the  common  difference,  I  the 
last  term,  n  the  number  of  terms,  and  s  the  sum  of  that  por- 
tion of  the  A.  P.  under  immediate  consideration. 

It  is  obvious  that  the  first  term  may  be  regarded  as  the 
last  and  the  last  term  as  the  first  according  to  choice. 

The  ^th  term  of  an  A.  P.,  considering  a  as  the  first,  is 

obviously  a-\-  Qn  —  1)  d.     Considering  the  nth.  term  as  the 

last  we  have  ,  ^        i^  -,  x^ n 

l  =  a+(^n-l}d.  (1) 

135.  The  sum  of  n  terms  may  easily  be  found  by  writing 
the  terms  in  two  forms.     Thus, 

s  =  a  -f  (a  -f  t7)  +  (a  +  2  c?)  -f-  •  •  •  +  (a  +  71—1  d) ; 

s  =  l+(l--d}-hCl- 2d} +'■■-{- (I-  /T^^l  d). 

95 


96  COLLEGE   ALGEBRA 

Adding,  we  have         2  8  =  na-\-nl, 
or,  s  =  |(«  +  0.  (2) 

To  find  the  sum  of  80  terms  of  the  series  of  even  numbers 
2,  4,  6,  •  •  • .     From  (1)  we  have 

Z=  2 +  (80 -1)2  =  160. 

and  from  (2)  we  have 

s  =  -822-(2  + 160)  =  6480. 

It  is  to  be  observed  that  (1)  and  (2)  are  independent  rehi- 
tions  between  the  five  quantities  a,  Z,  c?,  n,  s,  so  that,  given 
any  three  of  them,  we  may  find  the  other  two.  Thus  given 
a,  c?,  ^,  to  find  I  and  s  we  have 

I  =z  a  +  (^n  —  1)  d^ 
and  substituting  this  value  of  I  in  (2)  we  have 


n 


s  =  -(^a  -\-  a  +  n  —  1  '  tQ, 


n 


or,  s  =  -  (2  a  +  w  —  1  •  c?). 

EXAMPLES 

1.  Find  the  common  difference  and  sum  of  30  terms  of  an 
A.  P.,  the  first  term  of  which  is  6  and  the  hast  term  is  586. 

2.  The  sum  of  a  number  of  terms  of  an  A.  P.  is  220,  the 
common  difference  is  2,  and  the  last  term  is  31.  Find  the 
first  term  and  the  number  of  terms. 

3.  The  sum  of  16  terms  of  an  A.  P.  is  256,  and  the  last 
term  is  31.     Find  the  first  term  and  the  common  difference. 

4.  The  sum  of  5  terms  of  an  arithmetical  progression  is 
35.     Find  the  third  tei'm. 

Suggestion.     Let  the  series  be  x  —  2d,  x  —  d,  x,  x  -\-  d,  x  -\-  2d. 


PROGRESSIONS  97 

136.  Definitions.  Let  a,  A^  h  be  three  consecutive  numbers 
in  arithmetical  progression.  Then  A  is  called  the  arithmet- 
ieal  mean  between  a  and  h. 

By  definition,  A-a  =  h-A, 

or,  2  J.  =  a  +  5, 

A  =  --. 

In  the  same  way  if  a,  A^  B,  «7,  •  •  •,  5  are  consecutive  terms 
of  an  A.  P.,  then  ^,  J5,  (7,  •  •  •  aie  said  to  be  arithmetical  meayis 
between  a  and  b. 

137.  The  problem  to  insert  ru  arithmetical  means  between 

p  and  q  may  be  readily  solved,  for  we  have  the  first  term 

a—f^  the  last  term  I  =  g,  and  the  number  of  terms  n  —  m-\-2, 

so  that  7 

7      I  —  a  Q  —  P 

d  = -,  or,  ^— ^. 

^i  —  1  m  +  1 

Therefore  the  terms  are 

m  + 1  wi  + 1  7n-\-l 

EXAMPLES 

1.  Insert  5  arithmetical  means  between  2  and  16. 

2.  Insert  7  arithmetical  means  between  —  3  and  21. 

3.  The  sum  of  8  numbers  in  A.  P.  is  9  and  their  product 
is  15.     Find  the  numbers. 

Suggestion.     Let  x  —  d,  x,  x  -\-  d  be  the  numbers. 

4.  If  the  5th  term  of  an  A.  P.  is  7  and  the  12th  term  is 
50,  find  the  20th  term. 

5.  If  the  14th  term  of  an  A.  P.  is  —  2  and  the  19th  term 
is  16,  find  the  21st  term. 


98  X  COLLEGE   ALGEBRA 

6.  If  the  20th  term  of  an  A.  P.  is  —  15  and  the  4th  term 
is  4,  find  the  12th. 

7.  The  population  of  a  certain  town  increases  annually  in 
A.  P.  In  1892  it  was  1023;  in  1900  it  was  1175.  What 
was  its  population  in  1888  ?  What  is  it  in  1907  ?  What  will 
it  be  in  1921? 

8.  It  has  been  shown  in  mechanics  that,  if  the  resistance 
of  the  atmosphere  be  neglected,  a  falling  body  moves  over  a 
distance  of  16.1  ft.  in  the  first  second  and  48.3  ft.  in  the  sec- 
ond second,  and  so  on  in  arit/.imetical  progression.  How  far 
will  it  fall  in  10  seconds  ? 

9.  A  body  falls  ^g  ft.  in  the  first  second  and  |  g  ft.  in  the 
second  and  so  on  in  A.  P.     How  far  does  it  fall  in  t  seconds? 

Note.  The  law  expressed  in  example  9  still  holds  when  t  is  not  an 
integer,  but  the  proof  of  this  is  obtained  by  means  of  the  calculus. 

10.  Assuming  the  conditions  of  example  8,  how  long  would 
it  take  the  body  to  fall  788.9  ft.? 

11.  A  farmer  is  to  build  a  fence  825  ft.  long  with  the  posts 
one  rod  apart.  He  must  carry  these  one  at  a  time  from  one 
end  of  the  line.  How  far  will  he  have  traveled  when  the 
posts  are  all  distributed  and  he  has  returned  to  the  starting 
point? 

12.  The  three  digits  of  a  number  are  in  A.  P.,  their  sum 
is  12,  and  the  last  digit  exceeds  the  first  by  the  amount  of 
the  middle  digit.     Find  the  number. 

13.  A  number  consists  of  three  digits  in  A.  P.  Their  sum 
is  15  and  their  product  is  105.     Find  the  number. 

14.  A  number  consists  of  4  digits  in  A.  P.  Their  sum  is 
20.  The  product  of  the  first  and  fourth  is  to  the  product  of 
the  second  and  third  as  2  is  to  3.     Find  the  number. 


PKOGRESSIONS  99 

15.  If  «^  h\  c^  are  in  A.  P.,  then ,  , are  in 

.p  b  -\-  c    c  ■\-  a    a-\-o 

16.  Show  that  the  sum  of  any  2  7i  +  l  consecutive  integers 
is  divisible  by  2/1  +  1. 

17.  Show  that  the  sum  of  any  2  n  consecutive  integers  is 
not  divisible  by  2  n. 

18.  The  sum  of  any  number  of  consecutive  odd  integers  be- 
ginning with  1,  i.e.  1,  3,  5,  •  •  •  is  a  perfect  square. 

19.  If  2n-\-l  terms  of  the  series  of  the  last  example  be 
taken,  then  the  sum  of  the  alternate  terms  1,  5,  9,  •  •  •  is  to 
the  sum  of  the  remaining  terms  3,  7,  11,  •  •  ■  'ds  n  -{- 1  is  to  n. 


GEOMETRICAL  PROGRESSION 

138.  Definition.  A  geometrical  progression  (G.  P.)  is  a  suc- 
cession of  terms  such  that  the  ratio  of  any  one  to  the  preceding 
is  constant  throughout  the  progression. 

It  is  thus  seen  that  each  term  is  formed  from  the  preced- 
ing by  multiplying  by  this  constant  ratio. 

Example.  Show  that  the  terms  of  a  G.P.  form  a  con- 
tinued proportion. 

Here,  as  in  arithmetical  progression,  the  entire  number  of 
terms  is  unlimited. 

Let  a  denote  the  first  term,  r  the  constant  ratio,  I  the  last 
term,  s  the  sum  of  the  terms,  and  n  the  number  of  terms  of 
that  portion  of  the  progression  under  consideration. 

The  nth.  term  of  the  progression  considering  a  as  the  first 

term  is  obviously  ar""i.     Considering  the  nxh.  term  as  the 

last, 

l=ar^-^.  (1) 


100  COLLEGE  ALGEBRA 

139.    The  sum  of  n  terms  can  readily  be  found  as  follows  : 

s=  a-i-  ar  -f  ar^ H-  •  •  •  +  ar""^  -\-  ar'^~^^ 
rs  =  ar  +  ar^  +  ar^  +  •  •  •  +  ^r"~i  +  ar", 

or  subtracting  we  have, 

s  —  sr=  a  —  ar^^ 

or  s(l  —  r)=  a(l— r'^), 

or  «=— ^ -'  (2) 

1  —  r 

This  formula  for  the  sum  may  also  be  obtained  as  follows: 

s  =  a  +  ar  +  ar^  +  •  •  •  +  ar^~^ 
=  a(l  +  rH-r2+  ■••  +  r"-i) 

^'i^l^n.  (106) 

1  ~  r 

Example  :  To  find  the  sum  of  6  terms  of  the  G.P.,  1,  2, 

4,  •••wehave  .-,      ^ex 

8=1^=^  =  63. 
1-2 

It  is  to  be  observed  that  (1)  and  (2)  are  independent 
relations,  and  that  given  any  three  of  the  five  quantities 
a,  Z,  r,  72,  s,  we  can  determine  the  other  two. 

Thus,  given  a,  Z,  w,  to  find  r  and  s.     From  (1)  we  have 

^  a 

and  from  (2)  we  have  _  7""i[^ 

s= — • 


PROGRESSIONS  101 

EXAMPLES 

1.  The  first  term  of  a  G.P.  is  2,  the  ratio  is  3.  Find  the 
fifth  term. 

2.  If  the  first  term  of  a  G.P.  is  2,  the  last  term  162,  and 
the  ratio  3,  find  the  sum  of  the  terms. 

140.  If  a,   (7,  h  are  three  consecutive  terms  in  a  G.P., 

then  G-  is  called  the  geometrical  mean  between  a  and  h. 

Since  by  definition         ^       , 

^— A 
a       (r' 

a'^=ah, 
or  '  G-  =  ±  ^ah. 

If  a,  (r,  H^  /,  •••,5  are  consecutive  terms  in  a  G.P.,  then 
G-^  H^  I^  •••  are  called  geometrical  means  between  a  and  h. 

The  problem  of  finding  m  geometrical  means  between  a 
and  h  is  reduced  to  that  of  finding  the  constant  ratio,  and 
this  has  already  been  done  in  139. 

Example.    Insert  5  geometrical  means  between  2  and  128. 
Here  the  number  of  terms  is  7,  so  that 

128  =  2  .  r6  or  r6  =  64  and  r  =  ±2. 

141.  EXAMPLES 

1.  Find  the  sum  of  six  terms  of  the  G.  P.  3,  —  2,  |,  •••. 

2.  Find  the  fifth  term  and  the  sum  of  five  terms  of  the 
G  P    12   9  -2_"L   ... 

3.  Insert  5  geometrical  means  between  2  and  3. 

4.  Given  I  =  1215,  a  =  b,  n=Q,  find  r  and  s. 

5.  Given  a  =  21,  1  =  5103,  r  =  3,  find  n  and  s. 

6.  Given  ?^  =  5,  r  =  ^,  ^  =  2,  find  a  and  s. 


102  COLLEGE  ALGEBRA 

7.  Given  n  =  4,  r=—  2,  ?  =  25,  find  a  and  s. 

^ .  7      8r  —  s  -\-  a     „      sr  —  s-\-a      Ir 

8.  Given  a,  r,  s,  prove  t  = ■ — ,  r"  = =  —  • 

r  a  a 

9.  Given  a=2,  r=  —  |,  s  =  -y^-,  find  I  and  ti. 

10.  The  sum  of  three  numbers  in  G.  P.  is  21,  and  their 
product  is  216.      Find  the  numbers. 

Suggestion.     Let  -,  x,  xy,  represent  the  numbers. 

y 

11.  The  sum  of  three  numbers  in  G.  P.  is  7  and  the  sum 
of  their  squares  is  91.     Find  the  numbers. 

12.  There  are  four  numbers  of  which  the  first  three  are 
in  G.  P.  and  the  last  three  in  A.  P.  Their  sum  is  13,  and  the 
third  minus  the  first  is  3.     Find  the  numbers. 

13.  If  ^,  ^,  c,  d  are  in  A. P.,  prove  that 

(a2  +  y^  +  c^)(hc  +  hd^  ed)  =  (P  +  c'^  +  d?){ah  +  hc^  ca). 

14.  Three  numbers  whose  sum  is  15  are  in  A.  P.  and  if 
1,  4, 19  be  added  to  them  respectively  the  results  are  in  G.  P. 
Find  the  numbers. 

142.  If  the  ratio  is  less  than  unity  the  sum  of  n  terms 
remains  finite  when   ?z  =  co.     Considering  the    formula   for 

the  sum,  we  have  „ 

s  = 


or 


1  —  r 
a  ar^ 


1  —  r      1  —  r 
ar'' 


in  which may  be  made  as  small  as  we  please  by  taking 

n  sufficiently  great,  since  r  is  less  than  unity. 
We  have  therefore 


s  = when  n  is  infinite. 

1  —  r 


PROGRESSIONS  103 

EXAMPLES 

1.  Find  the  sum  of  1  +  J  +  ;|  +  •••  to  infinity. 

2.  Find  the  sum  of  -^q  +  yl^  +  j-^q-q  +  •••to  infinity. 

3.  Find  the  value  of  0.29. 
Suggestion.     0.29  =  j%%  +  j-^'^  +  .-. 

4.  Find  the  value  of  0.325.  5.    Find  the  value  of  2.21. 

HARMONICAL  PROGRESSION 

143.  Definitions.  A  succession  of  terms  the  reciprocals  of 
which  are  consecutive  terms  of  an  arithmetical  progression  is 
called  a  harmonical  progression  (H.  P.). 

If  «,  H^  h  are  three  consecutive  terms  of  a  harmonical  progres- 
sion, then  H  is  called  the  harmonical  mean  between  a  and  h. 

By  definition  -,  — ,  -  are  three  consecutive  terms  of  an 
a    H    h 

1   l^\ 

A.  P.  so  that  == — - — , 

xz  2 

rr      2  ah 
or  H= 

a  +  5 

Again  if  a,  H,  K,  L^  -"^h  are  consecutive  terms  of  a  har- 
monical progression,  then  H^  K^  L^  •••  are  called  harmonical 
means  between  a  and  h. 

144.  The  arithmetical,  geometrical,  and  harmonical  means 
between  two  numbers  a  and  h  have  been  denoted  by  A^  (r,  H^ 
respectively,  and  the  relations 

2  ■:- 

a=^±^ab. 


104  COLLEGE   ALGEBRA 

have  been  obtained.     From  these  it  appears  that 


or  a  =  ±  VAR. 

That  is,  the  geometrical  mean  between  two  numbers  is  the 
geometrical  mean  between  the  arithmetical  and  harmonical 
means  of  the  same  two  numbers. 

From  the  relation  G  =  VAII  it  appears  that  (5^  lies  between 
A  and  R. 

145.  To  show  the  connection  between  the  harmonic  pro- 
gression of  algebra  and  the  harmonic  division  *.  of  a  line  in 
geometry,  let  the  line  AB  be  divided  harmonically  (in  the 
geometrical  sense)  at  P  and  P^  Denote  AP^  AB^  and  AP' 
by  x^  ?/,  and  z  respectively.  Then  from  the  harmonic  divi- 
sion of  the  line  we  have 


Fig.  25. 


or 


or 


or 


^-^_ 

^-y 

X 

z 

X      z 

y  = 

2xz 

x-\-z'' 

AB  = 

2AP- 

AP' 

AP^-AP' 

*  Problems  in  H.  P.  may  be  solved  by  reducing  them  to  the  corresponding 
problems  in  A.  P.,  but  the  general  foiTuula  for  the  sum  of  an  H.  P.  has  not 
been  found. 


PROGRESSIONS  105 

That  is,  when  a  line  AB  is  divided  harmonically  at  P  and 
P',  AB  is  the  algebraic  harmonic  mean  between  AP  and 
AP',  or  AP^  AB,  and  AP'  are  in  harmonical  progression ; 
and  conversely  if  AP,  AB,  and  AP'  are  in  harmonical 
progression,  AB  is  divided  harmonically  at  P  and  P' . 
This  shows  that  harmonic  progression  of  algebra  is  identical 
with  harmonic  division  of  geometry. 

146.  EXAMPLES 

1.  Find  the  tenth  term  of  the  harmonic  series,  J,  -|,  J,   •••• 

2.  Insert  five  harmonic  means  between  3  and  15. 

3.  If  A,  G-,  and  H  denote  the  arithmetic,  geometric,  and 
harmonic  means  between  two  positive  numbers  a  and  h, 
arrange  these  three  means  in  their  order  of  magnitude. 

4.  Find  the  harmonic  progression  of  which  the  eleventh 
term  is  \,  and  the  fifteenth  term  is  \. 

5.  Find  the  H.  P.  if  the  thirteenth  term  is  \  and  the 
nineteenth  term  is  \. 

6.  Find   the    seventh  term  of   the  harmonic  progression 

1-4-     15.     5      ... 

7.  If  2  is  a  harmonic  mean  between  a  and  h,  then 

1  +^  =  Ui 


z  —  a      z  —  b      a      h 

a 


8.    If  a,  h,  and  c  are  in  H.  P.  show  that 


.     TTT^  h  -\-  c      c  -\-  a      a-\-h 

are  m  H.  P. 

9.    If  a,  h,  and  c  are  in  H.  P.,  then  a,  a  —  c  and  a  —  h  are 
in  H.  P.     So  are  c,  c  —  a,  c—h  also  in  H.  P. 


106  COLLEGE   ALGEBRA 

147.  EXAMPLES 

1.  Find  the  12th  term  of  V2,  —  2,  2V2,  —  4,  etc. 

2.  On  the  ground  are  phiced  n  stones ;  the  distance  be- 
tween the  first  and  second  is  1  yd.,  between  the  second  and 
third  3  yd.,  between  the  third  and  fourth  5  yd.,  etc.  How 
far  would  a  person  have  to  travel  to  bring  them  one  by  one 
to  a  basket  placed  at  the  first  stone  ? 

3.  Find  the  sum  of  15  terms  of  3,  7,  11,   •••. 

4.  Find  the  sum  of  19  terms  of  1,  1|,  2,  2|-,   ••-. 

5.  Find  the  sum  of  7  terms  of  J,  \^  \^   •••. 

6.  Find  the  sum  of  10  terms  of  -^g^,  —  |,  -|,   •••. 

7.  Find  the  sum  to  infinity  of  |,  ||,  |^f,   •••. 

8.  Find  the  sum  to  infinity  of  —  |,  \^  —  |,  •••. 

9.  Find  the  value  of  0.127. 

10.  Find  the  value  of  0.245307. 

11.  Given  the  sum  of  the  series  21,  19,  17,  etc.,  to  n  terms 
to  be  120.     Find  the  last  term  and  the  value  of  n, 

12.  Divide  unity  into  four  parts  in  A.  P.  such  that  the 
sum  of  their  cubes  is  -^^. 

13.  Find  an  A.  P.  such  that  the  sum  of  the  first  five  terms 
is  one  fourth  of  the  sum  of  the  following  five  terms,  the  sum 
of  the  first  ten  terms  being  unity. 

14.  Find  the  series  of  arithmetical  means  between  1  and 
21  such  that  their  sum  has  to  the  sum  of  the  two  greatest 
of  them  the  ratio  11  : 4. 

15.  A  number  of  three  digits  in  A.  P.  is  equal  to  26  times 
the  sum  of  its  digits.  If  396  be  added  to  the  number,  the 
digits  are  reversed.     Find  the  number. 


PROGRESSIONS  107 

16.  Snm  to  infinity   V^  +  l^     L_,     1,    .... 

V2  -  1      2  -  V2      2 

17.  Show  that  the  square  root  of  0.444  is  0.666. 

18.  In  a  G.  P.  show  that  the  product  of  any  two  terms 
equidistant  from  a  given  term  is  always  the  same. 

19.  In  a  G.  P.  if  each  term  be  subtracted  from  the  suc- 
ceeding term,  tlie  differences  are  also  in  G.  P. 

20.  The  square  of  the  arithmetical  mean  of  two  quanti- 
ties is  equal  to  the  arithmetical  mean  of  the  arithmetical 
and  geometrical  means  of  the  squares  of  the  same  two 
quantities. 

21.  Prove  that  the  two  quantities  between  which  A  is  the 
arithmetical  and  Gr  the  geometrical  means  are  given  by  the 
formulas  ^  ^  ^/CA+ a)(A~  G). 

22.  Suppose  a  body  to  move  20  miles  the  first  minute, 
19  miles  the  second,  18  miles  the  third,  etc.  How  far  will 
it  go  before  it  stops  ? 

23.  Three  quantities  are  in  H.  P.  If  -|-  the  middle  term 
be  subtracted  from  each,  show  that  the  three  remainders  are 
in  G.  P. 

24.  Show  that  h^  is  greater  than,  equal  to,  or  less  than  ac 
according  as  a,  5,  e  are  in  A.  P.,  G.  P.,  or  H.  P. 


CHAPTER   VIII 
PERMUTATIONS   AND    COMBINATIONS 

148.  Definition.  A  permutation  is  a  group  of  some  or  all 
of  any  number  of  objects  taken  in  a  definite  order ^ 

149.  Definition.  A  combination  is  a  group  of  some  or  all 
of  any  number  of  objects  taken  without  reference  to  their  order. 

Definite  arrangement  characterizes  permutations,  while  a 
mere  selection  regardless  of  definite  arrangement  character- 
izes a  combination. 

Thus,  four  students  on  a  seat,  taken  as  a  group,  form  but 
one  combination :  but  they  could  evidently  be  arranged  in 
many  ways,  each  of  which  would  be  a  permutation.  Also 
any  three  of  the  four,  if  taken  as  a  group,  would  form  a 
single  combination  but  several  permutations. 

150.  Notation.  We  shall  denote  the  number  of  permutations 
of  n  different  objects  taken  r  at  a  time  by  „P,,,  and  the  number 
of  combinations  of  n  different  objects  take7i  r  at  a  time  by  „(7^. 

Other  notations  for  ^P^  are  "P,.,  P„^^,  and  for  „6',.,  "^OJ., 
O 

151.  Fundamental  Principle.  In  the  treatment  of  permu- 
tations and  combinations  the  following  fundamental  principle 
is  used  :  If  one  operation  can  be  performed  in  m  different  ways^ 
and  after  that  a  second  operation  can  be  performed  in  n  differ- 
ent ways  for  each  way  in  which  the  first  has  been  performed^  the 
number  of  different  ways  in  which  the  tivo  operations  can  be 

108 


PERMUTATIONS   AND   COMBINATIONS  109 

performed  together  is  mn.  For  with  the  first  way  of  per- 
forming the  first  operation  there  are  n  different  ways  of  per- 
forming the  second  and  thus  n  different  ways  of  performing 
the  two  together ;  with  the  second  way  of  performing  the 
first,  there  are  n  different  ways  of  performing  the  second, 
and  thus  n  other  different  ways  of  performing  the  two 
together.  Continuing  in  this  way,  it  is  seen  that  the  whole 
number  of  different  ways  of  performing  the  two  operations 
together  is  n  -\-  n  -f  •  -  -f  ?i  to  m  terms,  or  7nn.  In  the  same 
way  if  an  operation  can  be  performed  in  m  different  ways, 
and  after  that  a  second  in  n  different  ways,  and  after  that  a 
third  in  p  different  ways,  the  number  of  different  ways  of 
performing  all  three  together  is  mnp.  For  it  has  just  been 
shown  that  the  number  of  different  ways  of  performing  the 
first  two  together  is  mti^  and  hence  as  before  the  number  of 
different  ways  of  performing  these  two  and  the  third  is  mnp. 
And  in  general  if  an  operation  can  be  performed  in  m^  dif- 
ferent ways  and  afterwards  a  second  in  m^  different  ways,  •• 
up  to  an  nth  in  m,^  dift'erent  ways,  then  all  can  be  performed 
together  in  m^m^  •-•  m„  different  ways. 

Thus  if  a  traveler  can  go  from  Syracuse  to  New  York  in 

3  different  ways,  and  then  from  Ncay  York  to  Boston  in  4 
different  ways,  the  number  of  different  ways  in  which  he 
can  go  from  Syracuse  to  Boston  via  New  York  is  12. 

PERMUTATIONS 

152.  To  find  the  number  of  permutations  of  four  students 
taken  three  at  a  time.,  or^  what  is  the  same  thing.,  to  find  the 
number  of  different  arrangements  iyi  which  they  can  be  seated 
three  at  a  time  in  a  line. 

Of  the  3  seats  at  our  disposal  any  one  of  the  students  can 
occupy  the  first.     Therefore  the  first  seat  can  be  filled  in 

4  ways.     After  the  first  seat  has  been  filled,  any  one  of  the 


110  COLLEGE   ALGEBRA 

remaining  3  students  can  occupy  the  second  for  each  way 
in  which  the  first  has  been  filled,  and  therefore  the  two 
together  can  be  filled  in  4  x  3,  or  12,  ways.  After  the  first 
two  seats  have  been  filled,  either  of  the  remaining  two  students 
can  occupy  the  third,  and  therefore  the  3  seats  together  can 
be  filled  in  4  x  3  x  2,  or  24,  ways. 

153.  We  are  now  prepared  to  state  the  general  theorem : 
The  number  of  permutations  of  n  different  objects  taken  r  at  a 
time  {^P,)  is     ^(^^_i)(^^_2)(^_3)...(,,_^_l_l). 

Proof.  We  may  represent  the  objects  by  the  n  letters 
«j,  a^,  ^3,  •••,  a„.  This  problem  is  the  same  as  to  find  the 
number  of  different  ways  in  which  the  n  letters  can  be 
distributed  r  at  a  time  in  a  box  with  r  compartments,  one 
letter  in  each  compartment.  We  can  place  any  one  of  the 
n  letters  in  the  first  compartment,  thus  filling  it  in  n 
ways.  After  the  first  compartment  has  been  filled  we  can 
place  any  one  of  the  remaining  n  —  \  letters  in  the  second 
compartment  for  each  way  in  which  the  first  has  been  filled. 
Therefore  we  can  fill  the  two  together  in  n  (n  —  1)  different 
ways.  After  the  first  two  have  been  filled  we  can  place  any 
one  of  the  remaining  n  -•  2  letters  in  the  third  compartment 
for  each  way  in  which  the  first  two  have  been  filled.  There- 
fore we  can  fill  the  three  together  in  n  Qn  —  1)  (ji  —  2) 
different  ways. 

Reasoning  in  this  way,  and  noticing  that  for  each  com- 
partment filled  a  new  factor  is  introduced  which  is  one  less 
than  the  preceding  factor,  and  that  the  number  of  factors 
in  the  product  is  always  equal  to  the  number  of  compart- 
ments filled,  we  obtain  as  the  last  factor,  when  r  compart- 
ments are  to  be  filled,  /        i\  ,1 

'     n  —  (r  —  1)  ov  n  —  r  -[- 1, 

Therefore    „P^  =n(n-  r)(n  -  2)(7i  -  3)  ...  (w  -  r  +  1). 


PERMUTATIONS  AND   COMBINATIONS  111 

154.  Corollary.  If  all  the  objects  are  taJce7i  together^  r  =  n^ 
and  the  last  expression  becomes 

„P,  =  n(n-l)(n-2)...3.2.1. 

This  is  the  continued  product  obtained  by  taking  all  the 
natural  numbers  from  1  up  to  n,  inclusive,  and  is  called 
factorial  n,^  and  is  denoted  by  the  symbol  7il  or  \n.  We 
sh-all  employ  the  former  and  write 

155.  EXAMPLES 

1.    In  how  many  different  ways  can  six  people  walk  abreast  ? 

Beginning  at  one  end  of  the  line  any  one  of  the  six  can 
occupy  the  first  place,  and  any  one  of  the  remaining  five  the 
second  place ;  thus  the  first  and  second  places  can  together 

*  The  definition  of  factorial  n  as  given  above  applies  only  when  n 
is  a  positive  integer.     It,  however,  is  entirely  consistent  with  the  two 

equations  (w  —  1)  !  =  — ,  and  2  !  =  2,  since  n  !  =  n  •  (n  —  1)  \.     We  shall 

71 

use  these  two  equations  as  a  general  definition  of  factorial  n,  applying 
it  to  all  integral  values  of  n,  whether  positive,  negative,  or  zero.  The 
second  equation  is  necessary,  since  the  first  defines  only  the  ratio  between 
n!  and  (n  —  1)!,  and  some  other  statement  is  necessary  in  order  to  give 
the  absolute  value  of  n  !.  Starting  with  2  1  =  2,  the  form  n  !  =  n  •  (n  —  1)  ! 
will  enable  us  to  build  up  the  factorials  of  all  positive  integers ;  while 
the  form   (n  —  1)  !  =  —   will  give  the   factorials  for  the  lower  values 


of  n.     Thus 


1!  =  2]  =  1, 

9  ' 


0!  =  li  =  l, 

1 


(-l)!=:^=oo,etc., 

and  by  continuing  the  process  we  see  that  the  factorial  of  every  negative 
integer  is  infinity. 


112  COLLEGE  ALGEBRA 

be  occupied  in  6  x  5  ways ;  by  continuing  this  reasoning  we 
see  that  the  whole  number  of  ways  in  which  the  six  people 
can  walk  abreast  is  gPg  =6x5x4x3x2x1  =  720. 

2.  Given  eleven  trees.  In  how  many  different  ways  can 
a  row  of  three  of  them  be  set  out  ? 

Any  one  of  the  eleven  can  be  set  in  the  first  place,  and 
afterwards  any  one  of  the  remaining  ten  in  the  second,  and 
after  that  any  one  of  the  remaining  nine  in  the  third, 
and  thus  the  row  of  three  trees  can  be  set  in  11  x  10  x  9  =  990 
different  ways. 

3.  In  how  many  ways  can  eight  numbered  seats  about  a 
round  table  be  occupied  by  eight  people  ? 

Here  the  emphasis  is  placed,  not  upon  the  relative  posi- 
tions of  the  persons,  but  upon  the  different  occupation  of 
the  seats  by  the  persons.  Thus  the  first  seat  can  be  occu- 
pied by  any  one  of  the  8  persons,  the  second  by  any  one  of 
the  remaining  7,  and  hence  by  continuing  the  process  we 
obtain  gPg  as  the  number. 

4.  In  how  many  ways  can  eight  people  be  seated  with 
respect  to  each  other  about  a  round  table  ? 

Here  the  emphasis  is  not  upon  the  different  occupation 
of  the  seats,  but  upon  the  relative  positions  of  the  persons. 
Therefore  the  first  person  may  take  any  place  without  affect- 
ing the  arrangement ;  after  that  the  next  person  may  take 
any  one  of  7  seats,  the  next  any  one  of  6,  and  so  on,  and 
thus  the  number  of  ways  in  which  8  people  can  be  seated 
with  respect  to  each  other  is  ^P^- 

5.  In  how  many  ways  can  eight  numbered  seats  around  a 
table  be  occupied  by  four  gentleman  and  four  ladies,  so  that 
no  two  gentlemen  sit  side  by  side  ? 

The  first  can  be  occupied  by  any  one  of  the  8  persons, 
but  the  second  is  now  restricted  to  one  of  4,  the  next  is  re- 


PERMUTATIONS   AND   COMBINATIONS  113 

stricted  to  one  of  3,  and  so  on,  giving  8x4x3x3x2x2 
X  1  X  1  =  8  X  4P4  X  3P3  =  1152. 

6.  In  how  many  ways  can  four  gentlemen  and  four  ladies 
be  seated  with  respect  to  each  other  about  a  round  table  so 
that  no  two  gentlemen  sit  together  ? 

One  seat  may  be  occupied  without  affecting  the  order  of 

arrangement.     The  rest  can  be  occupied  in  ^P^  x  3P3  =  144 

ways. 

COMBINATIONS 

156.  Let  us  consider  the  number  of  combinations  of  5 
students  taken  3  at  a  time.  5C3  is  the  required  number  of 
combinations.  We  are  to  find  its  value.  Each  combination 
contains  3  students,  and  therefore  will  yield  3P3  =  6  permu- 
tations, and  therefore  ^6^3  combinations  will  yield  gC'g  x  6 
permutations.  But  this  is  the  whole  number  of  permuta- 
tions of  5  things  taken  3  at  a  time,  and  is  equal  to  5P3,  or  60. 

Hence  6X5(73=60. 

Therefore  5  Cg  =  10. 

157.  Let  us  now  take  up  the  general  theorem  of  which  the 
foregoing  is  a  special  example. 

Theorem.  The  number  of  comhinations  of  n  different  objects 
taken  r  at  a  time,  or  ^^C ,  is  equal  to  the  number 

7i(n—l)(n  —  2)  ■■•  (n  —  r-\-l) 
1.2.3--.r 
Peoof.  „C^  is  the  required  number  of  combinations. 
Each  of  the  „{7,.  combinations  contains  r  objects,  and  there- 
fore will  yield  j^P^  =  r\  permutations,  by  154.  Hence  „(7^ 
combinations  will  yield  „(7^  x  r !  permutations,  and  as  this  is 
the  whole  number  of  permutations  of  n  different  objects 
taken  r  at  a  time,  or  „P,,,  we  have 

nt,.  X  rl  =  nPr- 

I 


114 


COLLEGE   ALGEBRA 


Whence,  dividing  by  r !, 


n  _  n^r  _  ^(^-l)(^-2)  .♦•  (n-r+1)         .-  „. 


/* . 


Corollary  1. 
For,       „(7,= 

and 


1.2.3...r 


P  —   P 

T 

n(n—V)"-  {n  —  r-{-  2)' 
.        1.2--.(r-l)         _ 


(n  —  r-\-X) 


n      _  92  (n  —  1)  •  •  •  (y^  —  r  +  2) 
"    '-i~         1.2...rr-n 

Corollary  2. 

For,  _a  =  ^^^±^ 


n+1  '^r  —  •   n^r—Y 


n+1  ^r 


n  (?2  —  1)  •  •  •  (92  —  r  +  2)" 
(r-1)  ...2.1        ^. 


158.  Multiplying  both  numerator  and  denominator  of 
the  expression  found  for  ^Or  in  157  by  (?2  —  r)!  =  (/i  —  r) 
{n  —  r  —  X)  ••.3.2.1,  we  have 

r\{iii  —  r)\  ' 

where  the  numerator  is  now  seen  to  be  the  continued  product 
of  all  the  integers  from  7i  to  1,  or  yi !, 

/Ml 

and  therefore 


n  ___JL___ 
T\{n  —  r)\ 


As  between  this  formula  for  „(7^  and  that  of  157,  the  latter 
is  to  be  preferred  for  simply  calculating  „C^^,  though  an 
equivalent  formula  which  is  given  in  161  is  sometimes  to  be 
preferred  to  either,  while  the  formula  of  this  section  is  more 
compact  and  in  many  relations  is  to  be  preferred. 


PERMUTATIONS   AND   COMBINATIONS  115 

159.  Thus  far  the  symbol  „0q  has  no  meaning,  but  we  may 
define  it  as  the  value  which  the  above  expression  for  „C^ 
becomes  when  r  =  0,  that  is 

n^o  =  77T-H-  =  1-  (154,  footnote) 

160.  EXAMPLES 

1.  In  how  many  ways  can  the'  seats  of  a  four-oared  shell 
be  occupied  by  8  candidates  ? 

2.  In  how  many  ways  can  the  crew,  as  a  whole,  be  chosen 
in  the  preceding  example  ? 

It  is  to  be  noted  that  the  first  examplie  concerns  the 
arrangement  of  the  crew  in  their  seats,  the  second  only  the 
selection  of  the  crew. 

3.  In  how  many  ways  can  a  committee  consisting  of  3 
freshmen  and  4  sophomores  be  chosen  from  7  freshmen  and 
8  sophomores  ? 

The  3  freshmen  can  be  chosen  in  ^  C^  ways  ;  the  4  sopho- 
mores in  g6^4  ways  ;  each  choice  of  3  freshmen  can  be  com- 
bined with  each  choice  of  4  sophomores  to  form  a  committee  ; 
therefore  the  whole  number  of  possible  committees  is 

.a  x^a  =  — —  X  ^•'^•^•^  =  2450. 

^  ^     ^   *     1-2.3     1.2.3.4 

4.  From  7  different  roses,  6  different  carnations,  and  5 
different  chrysanthemums,  in  how  many  ways  can  2  roses, 
3  carnations,  and  1  chrysanthemum  be  chosen?      A71S.  2100. 

161.  Theorem.  The  nu7nber  of  combinations  of  n  different 
objects  taken  r  at  a  time  is  equal  to  the  yiiimber  of  combinations 
of  the  n  objects  taken  n  —  r  at  a  time^  orJJr  =  n^n-r-  For 
every  time  we  select  a  group  of  r  things,  there  remains  a 
corresponding  group  of  n  —  r  things.     Therefore  the  number 


116  COLLEGE   ALGEBRA 

of  groups  of  n  different  things  taken  r  at  a  time  is  equal  to 
the  number  of  groups  of  the  n  things  taken  7t  —  r  at  a  time. 
This  may  also  be  proved  as  follows : 

n  _  n\ 


r  !  ( >t  —  r)  ! 

n      _  "^  •  _  n\ 

(/i  —  r)  !  [n  —  (r^  —  r)J  !      {n  —  r)\  r\ 


and  therefore  nOr  =  n ^n-v 

When  n  objects  are  divided  into  two  groups  of  r  and  n  —  r 
respectively,  the  groups  so  formed  are  called  complementary 
combinations. 

The  relation  is  of  great  convenience  in  practice.  Thus  to 
find  the  value  of  53650  by  the  formula  given  in  157  would  be 
a  tedious  process,  but  instead  we  use  its  equal, 

,,(7,=  ^^- ^^-^^  =  23426. 

162.      Theorem.  n+l(^r  =  n^^r  +  n^r-V 

Pkoof.     We  have    „C^,.  =  „C^_i  •  ^~  •     (157,  Cor.  1) 

r 

Whence,  by  addition, 

=  »^.-i  •  {^)  =  ...1^.  (157,  Cor.  2) 

Another  Proof.  Let  us  set  aside  for  the  moment  one  of 
the  n-\-l  objects.  Then  the  number  of  combinations  of  the 
n  -\-l  objects  r  at  a  time  which  do  not  contain  this  object  is 
JJj. ;  and  every  combination  containing  it  must  contain  r  —  1 


TERMUTATIOXS   AND   COMBINATIONS  117 

of  the  other  n  objects,  and  so  there  will  be  n^r-i  of  them. 
But  the  two  classes  together  make  up  the  whole  number  of 
combinations  of  the  n  -\-l  objects  taken  r  at  a  time,  and 
therefore  n  _    n    ^     n 

The  latter  method  of  proof  leads  to  a  more  general  theorem 
of  which  the  above  is  a  special  case.  If  r,  s,  m,  and  7i  are 
positive  integers,  such  that  r  -\-  s  =  ???,  then 

For,  suppose  the  m  objects  to  be  divided  into  two  groups 
of  r  and  s  objects  respectively.  Then  „j(7„  includes  the 
combinations  for  which  the  entire  7i  come  from  the  first 
group,  those  for  which  7i—  1  come  from  that  group,  and  one 
from  the  second  group,  and  so  on  until  finally  it  includes 
those  for  which  all  come  from  the  second  group.     That  is, 

it  includes  ^C„,  r^n-i  '  s^v  ^"^^  ^'^  ^^^  ^^  s^n-  If  either  r 
or  s  is  less  than  n^  certain  terms  in  the  development  will 
vanish. 

If  in  this  development  r,  ?^,  and  s  become  w,  r,  and  1, 
respectively,  the  theorem  n+i^r  =  n^r-^  n-i^r  i^  obtained. 

163.    Let  us  now  examine  the  series 

P      n      P    ...      O    ...      P 

„v^Q,   n\-^Y>  71^2'         '  n^r'i         ">  n^n' 

»^.  =  ''~l^'^  ■  nO,.-v  (157,  Cor.  1) 

For  small  values  of  ?%  is  in  general  greater  than 

1,  but  as  r  increases  the  numerator  will  decrease  and  the 

denominator  will  increase,  and  thus will  decrease, 

r 

and  eventually  will  become  less  than  1.  So  long  as  it  is 
greater  than  1,  „(7^  will  be  greater  than  jfir-v  ^^^^  ^^^^  ^^^~ 


118  COLLEGE   ALGEBRA 

cessive  terms  of  the  series  will  be  growing  larger,  but  after 
becomes  less  than  1  they  will  be  growing  smaller. 

Let  us  find  the  value  of  r  for  which  ^O^  is  greatest. 

We  observe  that  there  are  n  +  1  terms  in  the  series,  and 
there  are  two  cases : 

First  when  n  is  odd^  and  hence  n-\-\  even.  In  accordance 
with  the  preceding  and  161  they  can  then  be  arranged  in 
pairs  of  equal  numbers  from  below  upwards  in  order  of 
ascending  magnitude  as  follows : 

O  —    C 


C   —     O 
fi  —     n 

Since  there  are  r  +  1  in  each  column,  and  n  +  1  all  together, 
we  have  o^     .  i\  .  i 

ryi    T 

therefore  r  = 


and  n  —  r  = 


2 

71  +  1 


2 

Therefore  when  n  is  odd  there  are  two  terms,  n^n-\  ^-nd 

n^n+v  which  are  equal  to  each  other  and  greater  than  any 

other  members  of  the  series. 

Second^  when  n  is  even^  and  hence  ?^  -f- 1  odd.  In  this  case, 
since  ^  -f- 1  is  odd,  the  greatest  term  of  the  series  is  left 
unpaired,  and  we  have  the  following  arrangement ; 


PERMUTATIONS  AND   COMBINATIONS  119 

C 

C        —     P 


o    —  o 

C        —      O 

n^O       —  n  ^n* 

Since  there  are  r  pairs, 

we  have  2  r  +  1  =  ?i  +  1, 

n 

and  J^n  is  the  greatest  term  of  the  series. 

2 

164.    Theorem.       Tlie   number  of  ways  of  dividing  m  +  n 
different  objects  into  tivo  groiqjs  of  m  and  /^,  respectively^  is 
(ni  +  7i) ! 
m\  n\ 

Suggestion.  This  is  equivalent  to  selecting  m  objects 
for  the  first  group.   See  158.   Let  the  student  supply  the  proof. 

Theorem.  The  number  of  ways  of  dividing  m  -^  n  -{-  p  differ- 
ent objects  into  three  grou2:>s  of  m,  n^  and  p^  respectively,  is 
(m  -\-  n+  p)\ 

ml  n\ p\ 

Proof.  The  group  of  m  objects  can  be  selected  from  the 
m-\-n-\- p  in  m+n+p(^m  ways ;  and  for  each  way  in  which  they 
are  selected  the  group  of  n  objects  can  be  chosen  from  the 
remaining  n-\-p  in  n+p^n  ways  and  therefore  the  two  selec- 
tions can  be  made  in 

n    V       p  _<J^i-^n-\-  py.      (n-\-py.      (m  +  n-{-py. 

m\{n+  p)\  n\ p\  m\  n\ pi 

ways. 


120  COLLEGE   ALGEBRA 

165.  Theorem.  The  number  of  j^armutations  of  n  objects,  k 
of  which  are  of  one  hind,  I  of  another  kind^  and  m  of  a  third 
kind,  and  the  rest  all  different,  is  equal  to 

nl 
k\  l\  m\ 

Proof.  Let  x  represent  the  required  number.  If  the  k 
like  things  be  replaced  by  k  unlike  things,  each  of  the  x  per- 
mutations will  yield  k !  permutations,  and  therefore  x  of 
them  will  yield  x  -  k\  permutations.  If  next  in  each  of  the 
X  '  k\  permutations  the  I  like  things  be  replaced  by  I  un- 
like things,  we  shall  similarly  obtain  x  •  k\  l\  permutations. 
Finally,  if  in  each  of  these  the  m  like  things  be  replaced  by 
m  which  are  unlike,  we  shall  have  x  -  k\l\  m\  permutations 
of  n  differeyit  things  taken  all  together.  But  this  number  is 
equal  to  n  !  by  154. 
Therefore  X'k\l\m\=n\, 

^^^  ^^TTT} — V 

kl  1 1  ml 

166.  Theorem.  The  number  of  permutations  of  n  different 
things  taken  r  at  a  time,  when  any  one  of  them  can  occur  once, 
twice,  and  so  on,  up  to  r  times  in  each  permutation,  is  equal  to  n^. 

Proof.  The  first  place  may  be  filled  in  n  ways,  and  the 
second  can  also  be  filled  in  n  ways,  since  each  thing  may  be 
used  more  than  once,  and  hence  the  two  together  can  be  filled 
in  nxn  =  n^  ways ;  in  the  same  way  each  place  may  be  filled 
in  n  ways,  and  therefore  the  number  of  ways  in  which  the  r 
places  may  be  together  filled  is  nxnxn  ■••  to  r  factors,  or  w'". 

167.  Theorem.  The  total  number  of  combinations  of  n  things 
taken  one  at  a  time,  two  at  a  time,  a7id  so  on  up  to  n  at  a  time, 
that  is,  />    I     /^    I     /H'    I  I    n 

n^l~r  n^2~^  n^3~r    '"   -Tn^n'' 

is  equal  to  2"  —  1. 


PERMUTATIONS   AND   COMBINATIONS  121 

Proof.  This  total  number  of  combinations  includes  the 
number  of  all  possible  ways  in  which  a  selection  may  be 
made  by  taking  some  or  all  of  the  things  at  a  time.  Hence 
in  making  the  selection  each  of  the  objects  can  be  treated  in 
either  of  two  ways  :  it  can  be  chosen  or  rejected.  Hence 
the  total  number  of  ways  of  treating  all  the  objects  is 
2  X  2  X  •••to  n  factors,  or  2"  ways.  But  this  includes  the 
case  in  which  all  the  objects  are  rejected,  and  the  number 
of  actual  selections,  or 

is  equal  to  2"  —  1. 

Another  proof  of  this  is  given  in  181,  Cor. 

168.  EXAMPLES 

1.  How  many  different  selections  of  5  coins  can  be  made 
from  a  bag  containing  an  eagle,  a  dollar,  a  dime,  a  sovereign, 
a  shilling,  a  mark,  a  franc,  a  lire,  a  gulden,  a  krone,  a  cent, 
a  pfennig,  and  a  farthing  ? 

2.  How  many  numbers  can  be  made  using  all  of  the 
digits  3,  5,  7,  8  ? 

3.  How  many  numbers  over  6000  can  be  made  with  the 
digits  3,  5,  7,  8  ? 

4.  How  many  different  arrangements  can  be  made  with 
the  letters  of  the  word  decimal  when  the  vowels  occupy  the 
even  places  ? 

5.  How  many  numbers  of  8  figures  can  be  made  with 
the  digits  1,  2,  3,  4,  5,  6,  7,  8  when  the  odd  and  even  digits 
alternate  ? 

6.  How  many  numbers  of  8  figures  can  be  made  with  the 
digits  0,  1,  2,  3,  4,  5,  6,  7  when  the  odd  and  even  digits 
alternate,  zero  being  considered  as  even  ? 


122  COLLEGE   ALGEBRA 

7.  If  the  number  of  permutations  of  n  things  taken  4  at 
a  time  is  20  times  the  number  of  permutations  of  ti  —  2 
things  taken  3  at  a  time,  find  n, 

8.  From  10  classical  and  8  philosophical  students,  how 
many  different  committees  can  be  formed  each  containing 

4  classical  and  3  philosophical  students  ? 

9.  If  15(7^  =  igO;.^,  find  ^(78,  and  ^^.^r-v 

Note.  —  In  problems  10,  11,  15  a  word  is  understood  to  mean  a  suc- 
cession of  letters. 

10.  Out  of  the  26  letters  of  the  alphabet,  in  how  many 
ways  can  a  word  be  made  consisting  of  5  different  letters, 
2  of  which  must  be  a  and  e  ? 

11.  How  many  words  can  be  formed  by  taking  3  consonants 
and  2  vowels  from  an  alphabet  containing  21  consonants  and 

5  vowels  ? 

12.  A  stage  will  accommodate  5  passengers  on  each  side ; 
in  how  many  ways  can  10  persons  take  their  seats  when  2  of 
them  remain  always  upon  one  side  and  a  third  upon  the 
other  ? 

13.  How  many  different  arrangements  can  be  made  from 
all  the  letters  of  the  words  Mississippi  ?  Cincinnati  ?  Phila- 
delphia ? 

14.  How  many  different  numbers  of  8  figures  can  be 
formed  with  the  digits  2,  2,  3,  3,  3,  5,  7,  7  ?  How  many  of 
9  figures  with  the  digits  1,  1,  4,  4,  4,  0,  0,  6,  6  ? 

15.  How  many  words  can  be  formed  from  the  letters  of 
the  word  Onondaga,  so  that  the  vowels  and  consonants  occur 
alternately  in  each  word  ? 

16.  A  war  vessel  has  a  signaling  system  of  five  colored 
electric  lights ;  each  color  has  3  distinct  positions.  Find  the 
total  number  of  signals  that  can  be  used. 


PERMUTATIONS   AND   COMBINATIONS  123 

17.  In  how  many  ways  can  n  things  be  given  to  p  persons 
when  there  is  no  restriction  as  to  the  number  of  things  each 
may  receive  ? 

18.  How  many  numbers  of  4  figures  each  can  be  formed 
with  the  digits  2,  3,  5,  7,  8,  9  when  there  is  no  restriction  as 
to  the  number  of  times  a  digit  may  be  repeated  in  each 
number  ? 

19.  In  how  many  ways  may  a  sum  of  money  be  drawn 
from  the  bag  mentioned  in  example  1  ? 


CHAPTER   IX 

BINOMIAL   THEOREM 

169.  In  this  chapter  we  shall  obtain  a  development  for 
the  7ith  power  (when  n  is  ?i  positive  integer)  of  a  binomial 
a-\-  b,  and  show  that, 

+  „(7X-^5^'+  ...+,(7,5^  (1) 

By  actual  multiplication  we  obtain  : 
(a-\-hy  =  a^-\-2ah-\-b^  =  ^CQa:^  +  ^C^ah  +  ^aj)%  (159,  157) 

(a  +  ^)3  =  a3  +  3  a2J  +  3  a J2  +  ^3  =  3  C^o^^H  3  C^i^^^  +  3  C>52  +  3  (7353, 

(^a  +  by=a^-\-4:a%-{-Qa^'^-{-4:ah^-{-b^ 

We  shall  now  show  that  these  developments  can  be  extended 
and  generalized. 

Let  n  be  any  value  for  which  we  have  verified  that 

(a  +  by=  ,  6>"  +  ,  O^a^-'b  +  . .  •  +  n  C,a"-''5'-  +...+,  C.,b\ 

and  let  us  see  what  will  follow. 

Multiply  both  members  of  this  equation  by  a-\-b.  We 
have  (first  multiplying  by  a  and  then  by  b  and  adding  and 

124 


BINOMIAL   THEOREM  125 

remembering  that  ,1^0  =  n+\0(^^  and  „C'„  =  n+i^«+n  since  each 
is  equal  to  one,  and  that 

n^r  "i"  n^r-l  =  n+V-'ri  (159,  162}) 


+  72^0 


«-^-M5'-+...+„(7„6'^+i 


or  (a-\-by"+^ 

Therefore  (a  +  ^)„+i  has,  when  expanded,  exactly  the  same 
form  with  respect  to  7i  +  1  that  (a  +  h~)^  had  with  respect  to 
n ;  and  therefore  the  development  is  true  for  a  value  of  n 
one  greater.  Since  we  have  verified  it  for  n  =  4,  it  is  true 
by  the  above  proof  for  n  =  5  ;  and  since  it  is  true  for  n  =  5 
it  is  true  for  n  =  6,  and  so  on  without  limit.  Hence  it  is 
true  universally  for  a  positive  integral  value  of  n. 

Remark.  The  form  of  proof  here  used  is  known  as  Mathematical  Induc- 
tion. It  consists  of  the  following  steps.  By  trial  in  a  few  cases,  which  may- 
be called  the  first,  second,  third,  etc.,  cases,  we  may  suppose  that  we  have 
discovered  a  law.  It  is  next  assumed  that  this  law  holds  for  the  wth  case 
in  order  to  build  up,  on  this  assumption,  the  {n  +  l)th  case.  If  the  (n  +  l)th 
case  has  the  same  form  with  respect  to  w  +  1  that  the  «th  had  with  respect 
to  w,  it  can  be  concluded  that  if  the  law  holds  for  the  ?ith  case,  that  is,  if  the 
assumption  was  true,  it  holds  for  one  more  case.  But  by  actual  calculation 
the  law  was  seen  to  hold  for,  say,  the  third  case.  Hence  it  holds  for  the 
fourth,  therefore  for  the  fifth,  etc.,  universally. 

It  must  be  clearly  borne  in  mind  that  this  mathematical  induction  is  not 
the  induction  of  logic  and  the  physical  sciences,  but  that  it  yields  absolute 
certainty  and  universality  in  the  conclusion.  For  additional  work  on  this 
topic  see  352-356. 

170.  It  will  be  shown  later  that  a  development  similar 
to  the  above  holds  under  certain  circumstances  when  n  is 


126  COLLEGE   ALGEBRA 

not  a  positive  integer.  In  order  to  state  tlie  form  of  this 
development  we  introduce  a  new  notation,  viz., 

7i(^  — l)(y^  —  2)  •"  (n  — r  +  1)  _  AA  ^^^ 

1.2-3  ...  r  ~\rj  ^  ^ 

for  any  value  of  n  whatever.     The  numbers  defined  by  the 

symbol  [     )  for   any  given  value  of  n   whatsoever,  and  the 

different  values  of  r,  viz.,  0,  1,  2,  •••,  are  called  binomial 
coefficients.     It  will  be  noticed  that  if  7^  is  a  positive  integer 

and  r > n, (     ]  =  n^r'i    i^   ^  is  a   positive    integer  and  r':>n, 

(  ^  J  =  0 ;  if  n  is  not  a  positive  integer,  I    ]  can  never  become 

zero. 

171.  The  development  before  mentioned,  when  n  is  not  a 
positive  integer,  will,  provided  a  is  numerically  greater  than 
5,  be  proved  in  221  to  be 

(a  +  by  =  (^\i^  +  h^  a^-^b  +  (^^\  a^-%'^  +  •  •  • 

+  r^y^-^^*-  +  •••  to  infinity.  (3) 

This  is  the  general  development  of  (a  -f-  by\  of  which  our 
previous  form  (1)  is  a  special  case,  for  when  ?i  is  a  positive 
integer,  (3)  becomes  identical  with  (1)  and  takes  the  form 

(a  +  by  =  (^\a-  +  (tja-n^  +  Q)«"-262  +  ... 

Jr(''\a--'b^^-\-''-  +  ('''\b^'  (4) 

The  notation  just  introduced  will  hereafter  be  used. 


BINOMIAL   THEOREM  127 

172.  Attention  is  here  called  to  the  fact  that  corollaries 
1  and  2  of  157  and  the  theorem  of  162  may  be  restated  and 
generalized  with  the  aid  of  the  new  notation  as  follows : 


(:)= 


n    \n  —  r  -}- 1      Ai  +  l^      f    ti    \n  +  '\. 


r  —  IJ         r        '    \    r     J      \r  —IJ 


The  proof  is  identical  with  that  given  before,  n  now  being 
any  real  number  whatever.  Let  the  student  rewrite  the 
proof  with  these  symbols. 

173.  In  regard  to  developments  (3)  and  (4),  it  may  be 
observed  that : 

1.  In  both  developments  the  exponent  of  a  in  the  first 
term  is  n,  and  decreases  by  one  in  each  succeeding  term ; 
the  exponent  of  h  is  zero  in  the  first  term  and  increases  by 
one  in  each  succeeding  term;  the  sum  of  the  exponents  of 
a  and  b  in  any  term  is  always  7i. 

2.  The  coefficient  of  the  first  term  is  unity ;  that  of  the 

second  is  obtained  from  the  first  by  multiplying  it  by  - , 
that  of  the  third  by  multiplying  the  second  coefficient  by 

-— — ,  and  in  general  that  of  the  (r  +  l)th  by  multiplying 

ji f  I  "1 

the  rib.  coefficient  by  -^^-,  since 

r 

rj      \r  —  1/         r 

3.  If  in  (4)  we  replace  5  by  —  5  Qn  being  a  positive  in- 
teger), the  development  becomes 


128  COLLEGE   ALGEBRA 

Ca  -  by  =  (^y  +  Qa^-i(^-b)  +  g^a-2(_  ^)2  +  ... 

+  (''V"-^(-  hy  +  •••  +  (^\-  by 

and  we  see  that  the  sign  of  a  term  is  +  or  —  according  as 
b  occurs  to  an  even  or  an  odd  power  in  that  term. 

4.  The  sign  of  a  term  in  (3)  depends  jointly  on  the  sign 
of  b  and  the  numerical  coefficient,  and  can  be  determined  in 
any  particular  case  for  given  values  of  b  and  n. 

5.  Equations  (3)  and  (4)  assume  the  following  form  when 

(1  +  a;)^  =  1  +  nx  +  ^^^  ~^x'^+  ... 

2 ! 

^  n(n-  l)(n  -  2)  ■'•  (n  -  r -hi) ^r  , 
r ! 

174.  EXAMPLES 

Expand : 

1.     (a-\-by.  6.     (2:2-2  2/3)5. 

3\3 


2.    (a -by.  7.      2;2- 

1/ 


3.  (a +  3^)4.  8.    (x^  +  2a)4. 

4.  (a; -2^)6.  9.    fa:^  +  ^"^^ 


5.    (2  a- 5  5)3.  10.    (rt2  +  3?,)i  to  four  terms. 

11.    Vl  —  x"^  into  a  series  giving  four  terms. 


BINOMIAL   THEOREM  129 

THE   GENERAL  TERM   OF    (a  +  6)" 

175.  The    (r  +  l)th  term  in  (3)    or   (4)  is  ^^^"ja""^^^  or 

—5^ ^^-^^-— — — ^ ^^ ^a^  ^o"^  and  may  be  considered 

1  .  2  •  •  r  -^ 

the  general  term  from  which  each  of  the  other  terms  may  be 
obtained  by  giving  different  values  to  r.  It  may  be  ob- 
served that  the  number  of  the  term  is  always  one  more  than 

the  lower  number  in  the  symbol  [     J  • 

176.  Example.     To  find  the  coefficient  of  a;~2o  in  / -)  . 

/^\  V^      ^/ 

The  (r  +  l)th  term  of  (a  +  hy  is  I    ja"-''b\ 

x^              2 
Here  a  =  — ,  5  = -,  and  n  =  15,  therefore  the  (r  +  l)th 

term  of  g  -  ^''  is  (^^"^  gj    "■(-  |J.    In  this  expression 

^30-2r 

we  find  the  factors  containing  x  to  be  — - —  =  x^~^^ ;  but  r  is 
to  have  such  a  value  that  this  exponent  shall  be  —  20. 
.•.30-5r=-20,  5r  =  50,  r=10. 

Putting  this  value  of  r  in  the  preceding  expression  for  the 
(r+  l)th  term,  we  get 


15    14. 13    12. 11  .21O0: 


-20 


1  .  2  .  3  .  4  .  5  .  35 


1025024  ^_,o 


81 

1025024 


Therefore  the  required  coefficient  of  a:  ^o  is 


81 


130 


COLLEGE   ALGEBRA 


SOME   PROPERTIES   OF   BINOMIAL   COEFFICIENTS 
177.  Several  properties  of  binomial  coefficients  have  already 


been  given,  as 


n 

71+1 

r 

n  +  1 
r 


n 
r  —  1 

71 

r  —  1 

=  ( ")  + 

J  J 


n  —  r  —  1 
r 

n  +  1 


n 
r  —1 


(172) 


and  if  n  is  a  positive  integer 


n 

71  —  7' 


(161) 


178.  From  this  last  statement  we  derive  the  following 
theorem :  Iti  the  expansioTi  of  (a  +  ^)",  whsTi  n  is  a  positive 
i7iteger^  the  coefficients  of  terms  equally  distant  from  the  begin- 
ning and  the  end  are  equal ;  for^  by  the  principle  just  stated. 


n 
n 

n 

71  —  \J 


n 


n  —  r. 


179. 

Find : 


EXAMPLES 

1.  The  7th  term  of  {x  +  y)!^. 

2.  The  5th  term  of  (3  2:  +  2  yf, 

3.  The  4th  term  of  (1  -  xy. 

4.  The  1th  term  of  (1  —  x^~'^. 


BINOMIAL   THEOREM  131 

Find  the  (r  +  l)tli  term  of: 

5.  (1-xy.  7.  (1-xyK  9.  (1-xy^. 

6.    (1-xy.  8.    (l-a;)-2.  10.    (1  -  x^. 

Find  the  coefficient  of  : 

11.  x^^  in  the  expansion  of  (^^  +  2)^^. 

12.  x^  in  the  expansion  of  (a:^  -f-  2  xy^. 

13.  a;  in  the  expansion  of  (.-r^  —  ^— )    • 

14.  a:""^  in  the  expansion  of  ( ^ ^  ]    • 

/  1\12 

15.  The  term  independent  of  a;  in  i  2  a: ]   • 

SimjDlify : 

16.  (3  +  Vi)6  +  (3  -  V^/.     17.    (:r+ V3)5+(:r-V3)5. 

Find  the  middle  term  in : 

18.    (^  +  ^Y'-  19.    Cx-{-x-^y\  20.    (^'  +  ^^'" 

21.  Find  the  middle  terms  of  (  2  a  — 


aj 

22.  In  the  expansion  of  (^1 -\- x^^  the  coefficients  of  the 
(2  r  +  l)th  and  (r  -f-  5)th  terms  are  equal  ;   find  r. 

23.  Find  n  when  the  coefficients  of  the  14th  and  20th 
terms  of  (1  +  xy  are  equal. 

24.  Find  the  relation  between  r  and  n  in  order  that  the 
coefficients  of  the  (r  —  6)th  and  (3  r  4-  2)th  terms  of  (1  +  ^0^" 
may  be  equal. 


132  COLLEGE   ALGEBRA 

180.    The  greatest  coefficient  in  (a  +  6)",  when  ti  is  a  posi 
tive  integer,  is,  by  163,  \Z)  when  n  is  even,  and  l^\  =  (^^ 


when  n  is  odd,  the  number  of  the  term  being  ■^  +  h  when 

n  is  even,  and  when  n  is  odd,  two  terms,  the  ( — - — Jth  and 
^"'"     J  th,  having  equal  coefficients,  which  are  greater  than 


that  of  any  other  term. 

181.    Theorem.     The  sum  of  the  binomial  coefficients  when  n 
is  a  positive  integer  is  equal  to  2". 

Proof.     If  in  the  expansion  of 

we  put  a  and  h  each  equal  to  one,  the  right  member  becomes 
merely  the  sum  of  the  coefficients,  since  any  power  oi  a  oy  h 
becomes  equal  to  one,  while  the  left  member  becomes  2"  and 
expresses  the  value  of  the  sum ;  thus 


Corollary.     Since  f     J  =  1,  the  above  expression  gives 

a  theorem  in  combinations,  which  has  been  proved  in  167  in 
a  different  way. 


BINOMIAL   THEOREM  133 

182.  Theorem.  When  n  is  a  jjositive  integer^  the  sum  of 
the  hinomial  coefficients  of  the  odd  terms  is  equal  to  the  sum  of 
those  of  the  even  terms. 

Proof.     If  we  put  a  =  1,  ^  =  —  1,  in  the  expansion  of 
ia  +  hy  =  (^^  a-  +  (^^  a-^h  +  •  •  •  +  (^\% 

we  obtain  ^      (n\     fn\  .  fn\     fn\  . 

"'■    ^HH> -^HH^- ■■■■ 

183.  Problem.     Find  the  greatest  term*  in  the  expansion  of 

(3  a;  +  4  yy  when  x  =  b  and  y  =  3. 

The  (r  +  l)th  term  or  t,^^  =  ("^V3  2^)7-^(4  ?/)% 

j^^7-r+l      4^^  (172) 

C  r  2tx 


,      _,8-r      12__,  32-4r 
r         lo  or 

*  By  the  greatest  term  is  meant  the  term  having  the  greatest  numerical 
value  independent  of  its  algebraic  sign.  To  find  the  numerically  greatest 
term  we  therefore  proceed  as  if  all  the  signs  were  positive.  Thus  the  numeri- 
cally greatest  term  in  (x  —  4i/)i9  when  x  =  2  and  y  =  3  is  the  same  as  the 
numerically  greatest  term  in  (x  +  4  yy^  for  the  same  values  of  x  and  y. 


134  COLLEGE   ALGEBRA 

32  —  4r 
and  ^^+1  >  ^,.,  while  r  increases  so  long  as >  1,  and  no 


5r 


longer.     That  is,  so  long  as 

32  —  4  r  >  5  r, 

32  >  9  r, 

r<3f. 

Therefore,  since  r  must  be  a  positive  integer,  the  greatest 
value  which  it  can  have  which  satisfies  the  inequality  32  — 
4r>5r  is  3.     Therefore  the  greatest  term  is  the  4th. 

EXPANSION   OF   A   MULTINOMIAL 

184.  The  expansion  of  a  multinomial  may  be  obtained  by 
grouping  the  terms  of  the  multinomial  into  the  form  of  a 
binomial.     Thus : 

=  (2;2- 3  0^)3+3(2:2-3  ^)24  +  3(2:3_ 3^)42.^43 

+  482^-144a: 

+64 

^a;6_9^5_^39^4_  99  a;^ + 156  x^- 144  x  +  64. 

To  obtain  the  general  term^  or  term  containing  a-^m^m^^ . . . 
aj'-,  in  the  expansion  of  (a'j  +  ^2  +  ^3+  •••  +  ^,.)",  we  observe 
that  (^a^  +  a^-\-  -•■  +«,.)'*=  (^^  +  ^2+  •••  +«,,)(aj  +  a2+  •••  +a^r) 
•••  (^a^  +  ^2  +  •••  +  a,.')  to  n  factors  and  is  homogeneous  of  de- 
gree n.     To  form  the  given  term  we  may  pick  out  a^  from 

Zj  of  the  n  factors  in  (     J  ways,  and  after  that  we  may  pick 

/    _  7  \ 
out  ^2  from  ?2  of  the  remaining  n—  l^  factors  in  (  1  ]  ways, 


BINOMIAL   THEOREM  135 

and  by  associating  all  the  ways  of  making  the  first  selection 
with  all  the  ways  of  making  the  second,  we  may  select  a^,  l^ 

times  and  a^,  I^  times  together  in  f     V  M  ways;  that  is, 

we  may  select  a^^^a^^-  in  r   j(  M  ways.     Similarly  from 

the  remaining  n—l-^  —  l^^  factors  we  may  select  a^  from    l^ 

factors  in  y~  ^~  2j  ways;  proceeding  in  this  way  we  see 

that  the  number  of  ways  in  which  the  product  a-lm^m^^  •••  a^^^ 
can  be  formed  is 

(n  —  I^  —  I^—-'-  —  Ir-iVi  _  nl 


n 


where  l^-\-l^-\-  •••  +  ?,.  =  n.     The  number  ' — —   is  the 

number  of  ways  in  which  the  product  a^m^m^m^* "-  a,!r 
appears  in  the  expansion  (^a^-\-a^-\-a^-{-  •"  -\- a^y\,  therefore 
it  is  the  coefficient  of  that  term,  and  lience  the  general  term 
in  the  expansion  of  (a^  +  a^-[-  a^-^  •••  +  <^;)"  is 

•a/i«o2  •••  a  Jr. 


To  find  the  coefficient  of  x''  in  (^q  +  a^x  +  <^2^^  +  •••  +  «r^0"* 
The  general  term  by  the  preceding  section  is 


n\ 


ft  '1*^'    •••  I/,-, , 


aQo(a-^xy^(a.-^x^y-.  •••  (a,^'')'»- 


136  COLLEGE   ALGEBRA 

Therefore  to  find  the  coefficient  of  x^  we  must  find  all  inte- 
gral solutions  satisfying  both  the  equations 

^1  +  2/2  +  3/34-  •••  -Vrl^.^h. 

To  every  solution  corresponds  a  term  in  2;^.  The  col- 
lected coefficient  of  x^  is  the  sum  of  all  the  partial  coefficients 
corresponding  to  the  several  solutions  of  both  equations. 

Example.     Find  the  coefficient  of  a;^  in  (\-{-x-^x^-^  x^~)^. 
We  have  to  solve  the  equations 

'0  ^~  ^1 "'"  2  ~l~  ^3  ^^  ^' 
?i  +  2  Z2  +  3  ?3  =  6, 

and  find  as  solutions  for  Z^,  ?j,  l^^  Z3,  and  the  corresponding 
partial  coefficients, 


^0' 

h 

^2' 

h- 

Partial  coefficients. 

0 

1, 

1, 

1 

3!              _g 

^1 

0!1!1!1! 

0 

0, 

3, 

0 

-      ^-       -1 

^•> 

0!0!8!0! 

1, 

0, 

0, 

2 

3!        _ 

-f    t   rx  1    rx  1   ^1    ^1 

the  sum  of  which  is  10.     Hence  10  is  the  coefficient  of  x, 

EXTRACTION  OF   A   ROOT 

185.  The  following  example  shows  how  the  root  of  a  num- 
ber may  often  be  extracted  with  advantage  by  means  of  the 
expansion  (3),  171. 


BINOMIAL   THEOREM  137 

-^/63  =  63^  =  (64  -  1)'  =  64*  - 164-<^  +  iilnl)  64-^^.. 
^  ^  6  2! 

6  '  25     72  *  2"*" 
=  2-  0.005208  -  0.0000839  ... 
=  1,9947577+  ... 
=  1.994758. 

186.  EXAMPLES 

Find  the  sum  of  the  coefficients  of: 

1.  (x  +  ^y^.         3.  (x-\-2i/y.         5.  (x  —  2'i/y. 

2.    (x-^y\  4.    (x-2ijy.  6.    (2a;  +  3^-y. 

7.  A  man  invites  9  friends  to  dinner  1  at  a  time,  2  at  a 
time,  and  so  on  up  to  9  at  a  time.  How  many  different 
parties  does  he  form? 

Find  the  numerically  greatest  term  in : 

8.  (^x  +  I/)",  when    x  =  2    and  ?/  =  3  ;     when   x  =  2    and 

9.  (2;  —  ^)^S  when  x  =S  and  ?/  =  4. 

/  1\15 

10.  ix^+-j   ,  when  x  =  ^  and  «/  =  3. 

11.  (a:  —  3  ?/)i^,  when  x=l  and  ?/  =  f . 

12.  (2^4-  5)20,  when  a  =  |. 

13.  (2  a  +  3  by\  when  a  =  2  and  ?>  =  5. 


138  COLLEGE  ALGEBRA 

Expand : 

14.  (x-^y  +  zf.  15.  {x^  +2x-^f, 

16.  Find  the  coefficient  of  x'^  in  (1  -\-  x  -\-  x'^  +  a:^)^. 

17.  Find  the  coefficient  of  a;^  in  (1  -{- ^  x  +  x^ -\- 2>  x^Y. 

18.  Find  the  coefficient  of  x^'^  in  (2  —x  +  x'^  -\-  x^y. 

Find  to  5  places  of  decimals  : 

19.  V^.  20.    a/80.  21.    a/125. 


CHAPTER   X 
CONSTANTS,   VARIABLES,   LIMITS 

187.  Definitions.  A  constant  *  is  a  number  which  during  a 
given  discussion  does  7iot  change  in  value. 

A  variable  is  a  number  which  under  the  conditions  of  the 
problem  may  assume  various  values  during  the  same  discussion. 

Thus  the  circumference  of  a  given  circle  is  a  constant, 
while  the  perimeter  of  a  regular  polygon  of  7i  sides  inscribed 
in  it  is  a  variable  dependent  upon  n. 

188.  Definition.  If  under  the  law  of  its  change  a  variable 
number  can  be  made  to  approach  as  7iear  as  we  please  to  some 
constant  number  tvithout  becoming  ec^ual  to  it^  the  constatit  is 
called  the  limit  of  the  variable. 

Let  us  use  the  symbol  ^  to  signify  "  approaches  as  its 
limit,"  and  Lx  =  a  to  signify  the  limit  of  x  is  a.  Thus  x  =  a 
is  read  "a;  approaches  a  as  its  limit." 

In  the  previous  illustration  the  circumference  of  the  circle 
is  the  limit  of  the  perimeter  of  the  regular  polygon,  as  the 
number  of  sides  becomes  indefinitely  great;  the  area  of  the 
circle  is  the  limit  of  the  area  of  the  polygon. 

189.  The  sum  of  the  geometric  series  1  +  -  +  -H is  a 

variable  whose  limit   is   2.       This   can  be   proved  by  con- 
sidering the  formula  for  the  sum  of  n  terms  of  a  geometrical 

*  Constants  are  sometimes  absolute,  such  as  2,  3,  •••,  and  sometimes 
general,  as  r,  7i,  •••. 

139 


140  COLLEGE   ALGEBRA 

(tCA    T^^  CI  CLT^ 

progression,    s  =  — ^- ^=i- .      As   n   becomes 

indefinitely  great,  s  is  a  variable  and  approaches as  its 

n  1  —  r 

CIV 

limit,  since approaches  zero  when  r  is  less  than  1. 

1  —  r 

Thus  1  +  -  +  -  4-  •  •  •  approaches =  2  as  its  limit. 

2  4  1—2 

The  same  can  be  shown  geometrically  as  follows :  Take  a 
line  AB 

A  I ! ' H— *— »B 

Fig.  26. 

2  inches  in  length.  Take  half  of  it,  then  half  of  the  re- 
mainder, and  continue  the  process,  always  taking  half  of 
each  succeeding  remainder,  and  the  sum  of  all  the  parts  so 
taken  will  represent  the  given  series.  This  sum  evidently 
approaches  as  near  to  2  inches  as  we  please,  but  under  the 
law  of  formation  cannot  reach  it. 

190.  We  state  here  for  convenience  the  following  familiar 
theorems  without  proof.  A  treatment  of  limits  together 
with  the  proof  of  these  theorems  is  found  in  357-370. 

1.  If  two  variables  are  constantly  equals  and  each  approaches 
a  limits  their  limits  are  equal. 

2.  The  limit  of  the  algebraic  sum,  of  a  finite  number  of  vari- 
ables is  equal  to  the  algebraic  sum  of  their  limits. 

3.  The  limit  of  the  product  of  a  finite  number  of  variables 
is  equal  to  the  product  of  their  limits.  The  limit  of  a  constant 
times  a  variable  is  the  constant  times  the  limit  of  the  variable. 

4.  The  limit  of  the  quotient  of  two  variables  is  equal  to  the 
quotient  of  their  limits. 

191.  In  the  following  examples,  which  illustrate  the  use 
of  limits,  it  is  to  be  understood  that  {f(x)}^^^  =  L\f(x)}. 

Ex.  1.    Find  L  2^'-g^  +  2 

x=f)  OX^  -{-i  X—  D 


CONSTANTS,   VARIABLES,   LIMITS  141 

By  4,  190,  we  have 

o   2      Q      ,  o        L  (22;2-3a:H-2) 

a;  =  0 

=  -?.  (by  2, 190) 

o 

Ex.  2.    Find  L  ^^-^^  +  ^. 
x=oo02:^  +  7  a;  — 5 

Dividing  both  numerator  and  denominator  of  the  fraction 

by  x^.  we  have  o      o 

2-'^  +  4 
^  2^_Z^^+_2^   ^  X     x^ 

a;=Qo  5a;2  4- 7  a;  —  5      a-^oo  r   ,  7       5 

0-1 ^ 

a:      X'' 

=  -.  (by  4  and  2,  190) 

5 

^^    g      ^  (2  2;+5)(3rg-7)(2-3a;) 
:r^o  (rc2+5)(2:z:-7) 

i(2a;+5)-i:  (3a;-7)-X  (2-3:r) 

=  £±0 x=o x=o (by  4  and  3,  190) 

L  (2;2+5).Z  (2a;- 7) 

a;  =  0  x  =  ^ 

_  O^X-'^X^)  =  2.  (by  2,  190) 

5(-7) 

,  J   (2a:  +  5)(3a:-7)(2-3a:) 

.t^  (a;2+5)(2a:-7) 

r2  4-^V3-IY?-3 


J  \ Xj      \ X  J     \X /^     Q 

Q?'j\  Xj 


142  COLLEGE   ALGEBRA 

by  dividing  numerator  and  denominator  by  a?^  distributed 
among  the  different  factors  according  to  their  degrees,  and 
by  taking  limits  according  to  4,  3,  and  2,  190,  as  in  Ex- 
ample 3. 


■r^„    c      J  (a:  +  o)  ^ j^ V        ^^ 

V        xj\        X      x^Jx 


=  oo 


(1)     (1)     (0) 

by  dividing  numerator  and  denominator  by  a;*,  the  highest 
power  of  X  found  in  either,  and  completing  as  in  Example  4. 

Ex.  6.    Find  the  value  of  ■  when  x  =  a. 

X  —  a 

If  we  put  X  equal  to  a  in  this  fraction,  it  assumes  the  form 

of  -,  which  is  called  an  indeterminate  form,  because  in  this 
0 

form  its  value  is  not  apparent.  Its  appearance  does  not 
show  that  it  has  no  definite  value,  but  that  we  have  not 
used  the  proper  method  to  obtain  that  value.     So  long  as 

x^a^ =  1,  therefore  its  value  does  not  depend  on  x^  and 

X—  a 

hence  it  is  unity  for  all  values  of  x. 

Ex.  7.    Find  the  value  of  (tpAl±3\      . 

If  we  put  a:  =  2,  the  fraction  assumes  the  form  of  - . 

We  give  three  solutions  of  this  problem. 

(1)   We  may  factor  numerator  and  denominator  and  obtain 

a;2_5^-|-6_(a;-2)(a:-3)_a;-3 

.2 

2  —  3      1 

which  for  a;  =  2  is  equal  to  =  - 

^  2-4      2 


^-__6a;+8      (a;-2)(2;-4)      x-\' 


CONSTANTS,   VARIABLES,   LIMITS  143 

^"■^  ^2- 6  re +  8^=2    ^=^2(a:-2)(2;-4)  2      2' 

by  3,  190. 

(3)  Another  solution  is  obtained  by  putting  x=  2-^  h  and 
taking  the  limit  when  ^  =  0  ;   since  as  a;  =  2,  A  =  0.     Then 


^Y2:2-52:  +  6\     ^(2  +  /02-5(2  +  70H-6^  ^   Ji^  - 
z^2W  -  6  a:  +  8y     A^o(2  +  hy  -  6(2  -f-  A)  +  8      h^oh^-2 


h 


h^^h' h-2     2* 

In  the  preceding  example  we  found  the  expression  -  repre- 

1      ^ 
sented  the  value  1,  in  this  example  it  represents  -,  and  it  is 

clear  that  in  general  its  value  depends  upon  the  manner  in 
which  it  has  arisen. 


Ex.  8.    Find  the  value  of  ^ V-^— v^  +  V^  -  2  a 

If  we  put  x=2a^  this  assumes  the  form  of  - .      We  shall 

use  two  methods  of  avoiding  the  difficulty.  First,  by  factor- 
ing. We  notice  that  we  can  separate  the  fraction  into  two 
parts  : 

VJ— "\/2~a        1     Va:—  2  a 
and  . 


Vrr^  —  4  a^  Va;2  —  4  a^ 


Since         Va:^—  4  ci^  =  ^{x  +  2  a){x  —  2  a) 


and  Va;— 2  a  =-^(V2:4- V2  a)(Va;  — V2  rt), 


144 


COLLEGE   ALGEBRA 


we  have 


V(V2:-V2a)2 


'^x  —  V2  a\        _   __ 
Va;2 -  4  a?)x=2a     \y\(x+2a') ( Vi4-  V2a) (Vx-  V2^)^=«=2« 


VVa;-V2 


a 


0 


=2a 


and 


-\Jx—  2  a 


V4a(2V2a) 


Va;  —  2  « 


0, 


V2;2  -  4  a2y^2a       V  V(aJ  +  2  a)  (2J  -  2  «)^^=2a        V4  a        2  Va ' 

-\fx  — '\/ 2  a -\- '\/ X  —  2  a 


hence 


Va;2  —  4  a^ 


^2a         2Va 


Second,  rewriting  the  original  fraction  with  fractional  ex- 
ponents, and   putting  a;  =  2  a  +  A,    and   noticing  that   when 

a:  =  2  «,  7i  =  0,  we  have 

■\/x--^I^^-\/x-2a\       ^/(2a+A)^-(2a)i  +  Ai 

/^2a        V  (4«A  +  A2-)i 


V, 


x'- 


4«2 


/i=0 


and  expanding  the  first  term  of  the  numerator  by  the  bino- 
mial theorem,  and  factoring  the  denominator,  we  have 


(2a)i+l(2a)-U4- (2a)^  +  A^ 


7i2(4a  + A)2 


^=0 


l(2a)-UH-  •••  +A^ 


A2(4a  +  /0^        j^=o 


CONSTANTS,  VARIABLES,   LIMITS 


145 


Dividing  both  numerator  and  denominator  by  A^,  we  have 


(2a)-^Ai+  .••  +1 


(4  a  +  hy 
Ex.  9.    Find  the  value  of 


^  /i=o      (4a)2      2V 


a 


( 


V«^  +  aa:  +  a;2  —  Va^  —  aa;  +  a;^ 


d:=0 


Multiplying  both  the  numerator  and  the  denominator  by  the 
conjugate  surd  of  each,  we  have 

2  ax{;\/ a  -\-x-\-  ^ a  —  x) 

2  2:(  Va^  -}- ax -\- x^ -\-  Va^  —  ax-\-  x^} 

as  the  value  of  the  fraction  for  any  value  of  x,  and  its  value 

2  a 


i^"  —  v' 


192.    To  find  the  value  of 

\  U—  V  Jv=u 

We  distinguish  three  cases  : 

1.  When  n  is  a  positive  integer. 

2.  When  n  is  2i  positive  fraction. 

3.  When  n  is  negative  whether  integral  or  fractional. 


1. 


W''  —  V 
U  —  V  Jv=u 


=  (i^"-lH-W"-2v  +  W«-3|;2+    ...    _|-t,«-l)^,^^ 


=  nu^  ^ 


2.    When  n=^.     Put  u'^  =  x,  v*^  =  y,  then 
9 


p       p 
u  —  v 


x*- 


r\    _ 


X"" 


1 L.-, 


yvy=. 


X^ 


y 


p^ 


r'l  —   )i'l 


_px 


•p-1 


qpC 


9-1 


146 


COLLEGE    ALGEBRA 


by  case  1.     Multiplying  both  numerator  and  denominator  by 
x^  we  have 


p       p 


U  —  V 


q    x'^      q     u       q 


3.    Put  n  =  —  m,  then 


71 


-m         ny—m 


^m  _  ^m 


II  —  V       Jv=u        \U'"V^'\U  —  Vyjv=u 


VIM 


m—l 


u- 
=  —  mu 


—  m—l 


We  see  that  in  each  case  we  obtain  a  result  of  the  same 
form  in  terms  of  n. 

193.  EXAMPLES 

Find  the  limits  when  a;  =  0,  and  when  x  =  cc^  of : 


\  —  x^     ^  1  —  X 


2a^-l        2 


x^ 


(^-x)(x-h^)(2-lx) 
(lx-l)(x  +  iy 


Find  the  limits  when  n  =  0,  and  when  n=  cc,  of : 


3. 


4. 


n 


n  —  1 
n  +  1 


n 


5. 


6. 


a  +  (ri  - 

-l^d 

a  +  (^n  - 

-2)d 
1 

^2        (n_l)2  (2n  +  l)(27i  +  2)      (27i-l){27iy 

x^-{-l 


7.    Find    L 


x^-l  X 


2-1 


8. 


Find  the  value  of  : 

\a^  —  .r^)^  ^(^a  —  xy 
{a^  —  x^y  +  (a  —  a:)2^^=« 


a^  4-  a;^  +  a:^  +  1 


2; 


-1 


x=0 


CHAPTER   XI 
SERIES 

194.  Definitions.  A  series  z'.s  a  succession  of  terms  each  of 
which  is  formed  according  to  the  same  law.  If  after  a  certain 
number  of  terms  the  series  comes  to  an  end,  it  is  called  ^finite 
series.  If  the  terms  continue  in  an  endless  succession,  it  is 
called  an  infinite  series.  The  binomial  series  when  ^  is  a 
positive  integer  is  an  example  of  a  finite  series.  When  n  is 
not  a  positive  integer,  the  series  becomes  an  infinite  series. 

If  a  series  is  finite,  the  sum  of  its  terms  is  manifestly  some 
finite  number.  But  if  a  series  is  infinite,  the  sum  of  the  first 
n  terms  as  n  increases  is  a  variable  dependent  on  the  value  of 
n^  and  in  the  case  of  different  series  will,  in  general,  approach 
different  limits. 

195.  Definitions.  A  convergent  series  is  a  series  in  which 
the  sum  of  the  first  n  terms  approaches  a  finite  and  determinate 
limit  however  great  the  value  of  n  may  be.  We  define  the  sum 
of  a  convergent  series  as  the  limit  which  the  sum  of  the  first  n 
terms  approaches  when  n  becomes  indefinitely  great.  Thus  the 
geometric  series  l  +  J  +  i  +  J+  •••  approaches  the  limit  2,  is 
therefore  convergent,  and  2  is  the  sum  of  the  series  (189). 

A  divergent  series  is  a  series  in  ivhich  the  sum  of  the  first  n 
terms  does  not  approach  a  finite  and  determinate  limit  as  n  he- 
comes  indefinitely  great. 

If  a  convergent  series,  some  of  ivhose  terms  are  negative.,  re- 
mains convergeiit  wheji  all  its  terms  are  made  positive^  the  series 
is  said  to  he  absolutely  convergent. 

U7 


148  COLLEGE  ALGEBRA 

196.  Theorem.  Beginning  with  the  nth  term  of  a  convergent 
series^  the  sum  of  any  number  of  consecutive  terms  approaches 
zero  as  a  limit  as  n  becomes  indefinitely  great. 

Proof.     Let  the  series  be  denoted  by 

the  sum  of  n  terms  by  aS'^,  and  the  remainder  after  n  terms 
by  En-     Then 

Hence  ^n+m        *^n-l  =  ^»  +  ^/i+l  +    •••    + '^n+m* 

By  hypothesis  L  tS^  =  S;  (195) 

n=oo 

therefore  L  (^S^^^^  —  S^-i)  =  S  —  S=0, 

and  therefore     L  (^^„-f  Un+i  +  •••  +  w„+„J  =  0, 

for  any  value  of  m. 

Corollary  1.     In  particular  when  m  =  0  we  have 

L  (i^O  =  0. 

n=oo 

That  is,  the  nth  term  of  a  convergent  series  approaches  zero  as 
a  limit  when  n  becomes  indefinitely  great.  It  is  to  be  carefully 
noted  that  this  is  a  necessary  but  not  a  sufficient  condition  for 
a  convergent  series. 

Corollary  2.  If  the  nth  term  of  a  series  does  not  approach 
zero  as  a  limit  whe^i  n  becomes  indefinitely  greats  the  series 
cannot  be  convergent  and  is  therefore  divergent. 


SERIES  149 

Corollary  3.  The  remainder  'after  any  numher  of  ter-ins 
approaches  zero  as  that  numher  of  terms  becomes  indefinitely 
great.     For  when  m  =  go  in  the  expression 

■^  y^n  ~r  ^n-\-\  +    *••    H~  ^«+my» 
n=oo 

we  have  the  remainder  after  n—1  terms,  and  we  have  shown 
that  this  expression  has  zero  as  its  limit. 

Corollary  4.  Conversely,  if  the  remainder  Rn  after  n  terms 
approaches  zero-  as  a  limit  as  7i  becomes  indefinitely  greats  the 
series  is  convergent.  Since  by  taking  n  sufficiently  large  the 
possible  further  increase  or  decrease  of  Sn  can  be  made  less 
than  any  assigned  quantity  however  small,  therefore  S^ 
approaches  a  finite  and  determinate  limit,  and  by  definition, 
195,  the  series  is  convergent. 

From  Corollaries  3  and  4  we  observe  that  the  condition 
that  R^  approaches  zero  as  its  limit  is  both  a  necessary  and 
a  sufficient  condition  for  a  convergent  series. 

197.  If  the  series  which  begins  with  a  given  term  of  a 
given  series  is  convergent,  the  entire  series  is  convergent  by 
the  definition  of  a  convergent  series,  and  conversely,  195. 

And  if  the  series  which  begins  with  any  term  of  a  given 
series  is  divergent,  the  entire  series  is  divergent  by  the 
definition  of  a  divergent  series,  and  conversely,  195. 

198.  If  a  series  consisting  of  positive  terms  is  convergent, 
any  series  consisting  of  positive  terms  which  are  as  small  as 
the  corresponding  terms  of  the  first  series,  or  a  series  formed 
from  either  by  taking  any  or  all  of  its  terms  with  negative 
signs,  is  convergent.  For  in  either  case  the  remainder  after 
n  terms  will  be  at  least  as  small  as  or  numerically  smaller 
than  that  of  the  first  series,  hence  this  remainder  will  also 
approach  zero  as  a  limit,  and  tli,eref9re  either. series  will  be 
convergent.  T"      \       "T^    V 

_-.  _   -  -\  -■■ 


150  COLLEGE   ALGEBRA 

199.  As  explained  in  120,  when  x  is  real,  the  symbol  \x\ 
denotes  the  numerical  or  absolute  value  of  x ;  thus,  |  —  2 1  =  2 
and  I  2 1  =  2. 

200.  We  shall  now  consider  the  convergence  of  the  geo- 
metric series     -too  -, 

l-i-x-\-x^-{-x^+  '•'  -i-x^'-^-h  .... 


By  division,  or  by  the  theory  of  the  sum  of  a  geometric 
series, 

l-{-x-\-x^-\-a^-\-  •••  4-  x^^~^  = 


l-x""  1  x" 


-  X  1  —  X         1  —  X 

There  are  three  cases  to  consider  according  as 

|2:|<1,   |a;|>l,  or  I  a;  1  =  1. 

First,  when  |a^|  <  1.  In  this  case  x^  =  0  as  n  becomes  in- 
definitely great.    The  denominator,  1  —  a;,  is  constant.    There- 

fore =  0.      Hence  the  limit  of  the  sum  of  n  terms  as  n 

1  —  X  -| 

becomes  indefinitely  great  is   ,  which  is   a  finite  and 

1  —  X 

determinate  limit.  Therefore  by  the  definition  of  a  con- 
vergent series,  195,  the  series  is  convergent. 

Second,  when  |a7|>l.  In  this  case  x"^  becomes  infinite  as 
n  becomes  indefinitely  great ;  and  therefore  the  sum  of  7i 
terms  does  not  approach  a  finite  limit,  and  therefore  the 
series  is  divergent,  by  195. 

Third,  when  x=  -\-l.  In  this  case  each  term  is  unity  and 
the  sum  of  n  terms  becomes  infinite  as  n  becomes  infinite, 
and  the  series  is  therefore  divergent. 

When  x=  —  1  the  sum  of  7i  terms  is  alternately  1  and  0, 
and  therefore  the  sum  of  n  terms  does  not  approach  a  deter- 
minate limit.     Therefore  the  series  is  by  definition  divergent. 

This  last  kind  of  divergent  series  is  called  an  oscillating 
series. 


SERIES  151 

201.  Definition.  An  oscillating  series  is  a  divergent  series 
in  which  the  sum  of  n  terms,  though  always  finite^  does  not 
approach  a  deternmiate  limit. 

202.  The  question  of  the  convergence  or  divergence  of  a 
given  series  is  of  the  utmost  importance.  For  in  the  subject 
of  mathematical  physics  and  in  other  branches  of  applied 
mathematics  it  is  usually  necessary  to  throw  a  function  into 
the  form  of  a  series  in  order  to  calculate  its  value.  In  order 
that  the  series  may  correctly  represent  the  function,  it  is 
necessary  that  it  be  convergent.  The  danger  of  using  a 
divergent   series   to   represent   the   function   is   evident   by 

throwing  the  function into  the  form  of  a  series  by  actual 

division.     We  obtain 

=  1  +  2-  +  2;2  +  a.^  +  ...  -f  .T«  +  .... 


1-x 


We  have  just  shown  that  when  |.r|<l  this  series  is  con- 
vergent and  equal  to  the  first  member.  When  |  a:  |  >  1  we 
have  shown  that  it  is  divergent,  and  if  w^e  should  attempt  to 
use  a;  =  3  in  the  above  equation  we  should  have  the  absurdity 

-1  =  1  +  3  +  9+  ...  =00.      • 

Hence  the  physicist  first  examines  his  series  with  respect 
to  convergence  or  divergence  to  see  if  it  is  safe  to  use  it. 

METHODS  EOR  TESTING  THE  CONYERGENCY  OR  DIVERGENCY 

OF   A   SERIES 

203.  First  Method.  If  in  a  series  the  numerical  value  of 
each  term  is  greater  than  the  same  number  e  hoivever  small,  the 
series  is  divergent. 

The  proof  of  this  theorem  may  be  seen  by  observing  that 
it  is  a  special  case  of  failure  to  satisfy  the  conditions  of 


152  COLLEGE   ALGEBRA 

Corollary  1  of  196,  and  therefore  is  a  special  case  of  Corol- 
lary 2  of  196.  For  by  the  latter  corollary  any  series  what- 
ever, in  which  the  nth  term  does  not  approach  zero  as  a 
limit  as  n  becomes  indefinitely  great  must  be  divergent. 

Example.      Test  the  following  series  for  convergence 
or  divergence : 

5+^  +  5+.... 
5      7      9 

Here  we  observe  that  the  several  numerators  are  in  an 
arithmetical  progression  of  which  the  nth  term  is  n-{-  2; 
likewise  the  several  denominators  are  in  an  arithmetical 
progression  of  which  the  7ith  term  is  2  ?^  +  3.     Therefore  the 

n-{-  2 


nth  term  of  the  series,  u^  = 


2ri  +  3' 


But  .^^>;^  +  ^ 


2^+3     2^  +  4' 

that  is,  — ltj:L  >  _ 

2n-\-S     2' 

for  all  values. of  n;  therefore  the  series  is  divergent. 

204.  Second  Method.  A  series  of  alternately  positive  and 
negative  terms  m  which  each  term  is  7iumerically  smaller  than 
the  preceding^  and  in  which  the  nth  term  approaches  zero  as  a 
limits  is  convergent. 

Proof.     Let  the  series  be 

S  =  u^  —  n^  -\-u^  —  ?/^  +  •  •  • , 

then  we  have 

S  =(u^  —  u^  +  (u^  —  u^  +  (^^5  —  We)  H ,      (1) 

and  also       S  =  u^  —  (u^  —  ?^3)  —  (u,^  —  Wg)  —  •••,  (2) 


SERIES  153 

where  each  parenthesis,  in  itself,  represents  a  positive  number, 
since  by  hypothesis  each  term  is  numerically  less  than  the 
preceding. 

Form  (1)  represents  jS  as  the  sum  of  positive  numbers, 
and  shows  that  it  is  greater  than  Wj  —  ii^.  Form  (2)  shows 
that  S  is  less  than  Wj.  Therefore  S  is  positive  and  finite, 
and  lies  between  Wj  and  u^  —  u^. 

Again,  since 

we  have     |  i^„  |  =  Un+i  —  u,^+^  +  w^+g  —  u^^^  +  •  •  • , 

whence  |7^,J  can  be  thrown  into  the  two  forms  corresponding 
to  those  above  for  S. 

Thus,       \Bn\=  (w„+i  -  Un+2)  +  (^„4-3  -  %i+0  +  •  •  • ,  (3) 

and  also     \Rn\=  ^n+i  -  (i^n+2  -  '^^+3)  "  (^^«+4  "  w«+o)  ' •  '•     (4) 

Therefore  |i2„|  lies  between  \un+i\  and  |i*„+^  —  w^+gl*  ^^^ 
\Un+^\  and  \Un+i  —  Un+2\  botli  approach  zero  as  a  limit  when  n 
becomes  indefinitely  great.  Therefore  \Rn\  approaches  zero 
as  a  limit  when  n  becomes  indefinitely  great.  Therefore, 
by  196,  Cor.  4,  the  limit  is  determinate  and  the  series  is 
convergent. 

Example.      1 h-  —  -  +  - f-  •••• 

2      3      4      5      fc) 

Here  the  nth  term,  w,^  =  (— 1)"~^-.      Therefore  the  nth 

n 

term  approaches  zero  as  its  limit  as  n  becomes  indefinitely 
great,  each  term  is  numerically  smaller  than  the  preceding, 
and  the  terms  are  alternately  positive  and  negative.  Hence 
the  series  is  convergent. 


154  COLLEGE   ALGEBRA 

205.  Third  Method,  or  Method  of  Ratios.  This  method  will 
be  considered,  under  two  theorems: 

I.  Theorem.  If  m  an  infinite  series  heginning  with  and 
after  a  certain  term  the  ratio  of  each  term  to  the  preceding  is 
numerically  less  than  a  positive  number  which  is  itself  less  than 
1,  the  series  is  convergent. 

Proof.    First,  when  all  the  terms  of  the  series  are  positive. 
Let  the  series  beginning  with  this  certain  term  be  repre- 
sented by 

>S  =  Wj  +  2^2  +  ^^3  +  •••  +  ^r  +  •••• 

But    '^<k,'^<k,'^  <Tc,  ...,'^  <h,  ^^<k,  •••,-?;<l. 

U-^  U^  Uq  '^/i-2  '^^n-l 

Therefore,  by  multiplying  the  first  two  of  these  inequalities, 
then  the  first  three,  and  so  on,  we  have 

%     ^2   ^  1,2    ^4     ^3     ^2  ^  L3  '^^        ^^-1        ^4     ^3      '^h  ^  Z,n-1 

Un        U-l  Un        U(f        U-t  U^i_-l         Uf^_s)  11  o         Un  U-i 

or,  '^<k\  '^<k\  ...,  ^<F-i,  .-. 

Therefore,  by  clearing  of  fractions, 

U-\  —  U-i, 

Un    "^  U-irCtf 
Wg    <  U^'^^ 
U^   <  U-J^^ 

u''  <  u^V-\ 


SERIES  155 

that  is,  the  series  S  is  less,  term  by  term  (except  the  first), 
than  the  series 

jS'  =  u^  +  2i^k  +  u-Jc'^  -\-  ii^lc^  -\-  ...  -\-  u^k'^-'^  +  ••• 
=  ?/i(l  +  ^  +  A;2  +  F+...  +V-^+  ...)• 

But  the  series  1  -{-  k  +k'^  -{-  k^  +  k''~'^  +  ••-  approaches  a 
finite  and  determinate  limit,  by  200 ;   therefore  S'  =  Lim  SJ 

=  Lim  2^^  (1  +  y^  4-  A;2  +  F  +  . . .  +  A;"-i)  =  u^  (t^-t)  .  by  190,  3, 

is  finite  and  determinate.  Hence  S'  is  convergent,  195,  and 
JIJ  =  0,  196,  Cor.  3 ;  but  since  S  is  less,  term  by  term,  than 
aS^',  Rn<Rn^  and  so  much  the  more  B,,^  =  0  as  n  becomes 
indefinitely  great,  and  S  is  convergent,  by  196,  Cor.  4. 

Second.,  if  the  terms  of  S  are  not  all  positive,  the  series 
will  still  be  convergent,  by  198.  And  in  each  case  it  follows, 
by  197,  tliat  the  entire  series  is  convergent. 

II.  Theorem.  If  in  an  infinite  series^  beginning  with  and 
after  a  certain  term^  the  ratio  of  each  term  to  the  j^recediyig  is 
numerically  equal  to  or  greater  than  1,  the  series  is  divergent ; 
for  after  this  certain  term  the  terms  are  either  equal  or  in- 
creasing numerically,  therefore  the  nth.  term  cannot  approach 
zero  as  a  limit  Avhen  n  becomes  indefinitely  great,  and  by  196, 
Cor.  2,  the  series  is  divergent. 

206.    The  method  of  205  is  applied  in  practice  as  follows : 


If  the  limit  of 


u. 


^n-\ 


as  n  become  indefinitely  great  is  less 


than  1,  the  series  is  convergent,  for  the  hypothesis  of  Th.  I 
is  satisfied.  If  this  limit  is  greater  than  1,  the  series  is 
divergent,  for  the  hypothesis  of  Th.  II  is  satisfied.     If  the 

ratio   — —    is  greater  than  1  and  approaching  1  as  its  limit, 

the  series  is  also  divergent,  by  Th.  II.     But  if  the  ratio 


■«-i 


156  COLLEGE   ALGEBRA 

is  less  than  1  and  approaching  1  as  its  limit,  the  hypothesis 
of  Th.  I  is  not  satisfied,  and  therefore  we  cannot  decide  by 
this  method  whether  the  series  is  convergent  or  divergent. 
In  this  case  the  ratio  method  is  said  to  fail,  and  we  require 
another  test  for  such  series. 

17    .  3,3-4,  3-4.5  ,         ,    3-4.--(n+2)    , 

Example.     -  +  _-  +  -— -  +  ---  +  -— — /:,        .-,{  +  •••. 
4      4-b      4-6-8  4-b---(2^i  + 2) 

Here  u  .  _  3  •  4  ■  5  -  (»  + !)(»  + 2) 

Here  **»- 4  .  6  .  8  -  (2»)(2«+ 2)  ' 

from  which  Un_^  can  be  obtained  by  changing  n  into  n—1; 


thus,  Un-i 


3.4.5...  (n+1) 
4-6  -8..-  (2  n) 


Hence,  by  dividing  and  simplifying, 

u„  _  n-\-  2 
u^-i     2  /i  +  2 

Taking  the  limits,  X  -^  =  ^  •  (191) 

Therefore  the  series  is  convergent. 

207.  Fourth  Method,  or  Method  of  Comparison  of  Two 
Series. 

Theorem.  If  Si^=u^-{-  u^-\-  -"  +it„+  •••  and  S^,  =  v-^^-{-  v^ 
+  •••  +  ^M  +  •••?  ^^^  ^^^  series  coyisisting  of  positive  terms,  and  if 
the  ratio  — *  is  fiiiite  and  7iot  zero,  for  all  values  of  n  whatever, 

both  series  are  convergent  together,  or  else  divergent  together. 
Proof.    By  39,       B>    \    ^  ; , — ->r. 


SERIES  157 

where  R  and  r  are  the  greatest  and  the  least  values  of  the 
ratios    -\   ^^  ••.,  !^,     and    therefore    ^h  +  ^2  +  "•  +  ^"  ^  ^ 

where  x  is  finite  and  not  zero,  as  n  becomes  indefinitely 
great.  Whence  u^-\-  u^-\-  •  •  •  -\-iin  =  x(y^  +  ^2  +  " "  +  ^n)^  »^i^d 
by  taking  the  limits  of  both  members  of  this  equation  we 
obtain  8^  =  X  -  iS„,  where  JC  is  finite  and  not  zero,  from  which 
it  appears  that  both  series  are  convergent  together,  or  else 
divergent  together. 

208.  The  Auxiliary  Series.  For  purposes  of  comparison  in 
the  application  of  the  preceding  method  the  following  series 
is  of  great  advantage : 

Considering  its  convergency,  it  will  be  found  that  it  is  con- 
vergent if  r>l,  but  if  r>l,  it  is  divergent. 

Proof.     There  are  three  cases  according  as  r  is  greater 

than,  equal  to,  or  less  than  1. 

1.    In  the  given  series  the  sum  of  the  second   and  third 

2 
terms  is  less  than  — ,  the  sum  of  the  next  four  terms  is  less  than 

4  8 

— ,  the  sum  of  the  next  eight  terms  is  less  than  — ,  and  by 

^r  yr 

continuing  in  this  manner  we  see  that  the  sum  of  the  given 

series  is  less  than  ^        .       ^ 

l+lj-l-i-A-i-... 
2^     4^      8''         ' 

11  1 

that  is,  less  than  1  -\ -\ — -  -\ +  •  •  • . 

2''"^      ('2^~^)^      (2'~0 
But  the  latter  is  a  geometric  series  whose  ratio  is   —— ^   and 
when  r>l  this  ratio  is  less  than  1  and  the  geometric  series 


158  COLLEGE   ALGEBRA 

is  convergent,  by  200.     Therefore  so  much  the  more  is  the 
given  series  convergent  when  r>l  (198). 

2.    When  r  =  1,  the  series  becomes  the  harmonic  series^ 

1  +  1  +  U...  +  1  +  .... 

A      6  li 


In  this  series  the  sum  of  the  third  and  fourth  terms  is  greater 
than  -,  that  of  the  next  four  terms  is  greater  than  -,  likewise 
of  the  next  eight,  the  next  sixteen,  and  so  on.  Thus  the  sum 
of  the  series  is  greater  than  1  +  -H !--+•••  to    infinity, 

A       A       li 

therefore  the  series  is  divergent  (195). 

3.    When  r<l,    each  term   of  the  series  —  4- —  +  — H 

!'•      2'-      3'-      4^ 

+  •••  is  greater  than  the  corresponding  term  of  the  har- 
monic series,  and  therefore  the  series  is  divergent.  This  also 
includes  the  case  when  r  is  negative. 

209.  In  using  the  method  of  comparison  it  will  often  be 
found  convenient  to  take  the  series  v^  +  ^2  +  ^'3  +  '**  ^s  the 
auxiliary  series  — -\ 1 — ■4---..      To   determine  the  value 

of  r  that  is  to  be  chosen  in  any  particular  instance,  when  the 
terms  of  the  series  %i^  +  n^  +  W3  +  •  •  •  are  algebraic  fractions 
in  terms  of  w,  we  introduce  the  following  definition  and  rule. 

210.  Definition.  Tlie  total  degree  of  an  algebraic  fraction 
expressed  in  terms  of  n  is  the  degree  of  its  numerator  minus 
that   of  its    denominator   in   n.       Thus    the  total  degree    of 

~ — - —  is  2  —  3,  or  —  1.     The  total  degree  of  —     ^      is 

n^^b  Vn4  +  3 

The  total  degree  of  the  auxiliary  series  chosen  should  he  the 
same  as  that  of  the  series  to  he  tested.     Also  in  applying  the 


SERIES  159 

method  of  comparison,  we   consider  L  —•     If  this  limit  is 

71=00     n 

finite  and  not  zero,  the  series  are  both  convergent  together, 
or  else  both  divergent  together. 

Example.    Ll? +^a;+— ^i;^^  •••  _^^0^+ ^):^»-i  +  .. 
5         12  31  713  +  4 

X  being  considered  as  positive. 

Here       u„  =  ^ ^x^  S     w„_i  =  — ^^ f- — -x^  ^. 

71^  +  4:  "^      (^_  1)34. 4 

Hence  — n- —  l^ 1 — — ja — \ — /^^ 

Un-l  (^3+  4)0^-1) 

and  i  -^^  =  X,  (191) 

Therefore  if  x<l,  the  series  is  convergent,  (206) 

and  if  x>l,  it  is  divergent.  (206) 

But  if  2;=  1,  it  can  be  shown  that  — —  <  1  when  n>2,  and 

therefore  the  ratio  test  fails.  Let  us  then  use  the  method 
of  comparison  for  this  case.  The  total  degree  of  w„  is  —  1, 
and  hence  we  choose  for  auxiliary  series 

z      6      4:  n 

whose  nth.  term,  v.J^  =  -,  is  of  total  degree  —  1. 

n 

1  he  ratio  — ^  =  — ^-- — ' — -  • 

Vj^  ?r  +  4 

Hence  X  -^  =  1,  which  is  finite  and  not  zero.  Therefore 
both  series  are  convergent  together,  or  else  divergent  together. 


160  -  COLLEGE   ALGEBRA 

But  the  series  1  +  ^  +  o  "I —  ^^  divergent.  (208) 

Hence  the  series  — ^  +  ^^  +  -— —  +  •  •  •   +  ^'^^ — r^  +  •  •  •, 
5  12         31  n^-\-\ 

or  the  given  series  when  a;  =  1,  is  also  divergent.  (207) 

211.  In  testing  the  convergence  of  any  'particular  series^  the 
student  should  carefully  examine  the  various  methods  to  see 
which  applies  to  the  series  under  consideration.  When  more 
than  one  method  applies^  the  simplest  should  he  chosen. 

212.  EXAMPLES 

Examine  the  convergence  of  the  following  series : 

1  2     3     4      5 
2+4+6+8+      • 

2  1  +  3     5     1     17      ...    .  2''-'+l 

+  4  +  8  +  16  +  32+         +       2«      +     ■ 

3.  1_1  +  1_1  +  1_.... 

3     6     7     9 

4.  -i "—+       1 


a  +  5      a-\-2h      a  +  36 

33-5       3.5-7  3-5. T--.  (2^  +  1) 

4      4.7      4.7.10  4-  7.10...  (3 ^i  +  1) 

6.  ZxA \- h  •••• 

4  9         16        25 

7.  a-\-(^a-^d)x+  (^a-\-2d)x^+(ia  +  ^d)x^-\-  •'-. 

8.  — L_  +  _±_  +  ^^-|-  ...  -j ^"^ 1-.... 

3. 45. 67. 8  (2^  +  l)(2*i  +  2) 


SERIES 

9. 

i+   1   +   1   +  1   - 

X      x-\-\      x-\-2      x+S 

f.... 

10. 

I  +  -  +  2V3V-. 

12. 

1 

2!  '  4!       *"• 

11. 
14. 

X^     ,     X" 

*     3!  +  5!      •••• 
2  +  8       4       5       ^.._ 

13. 

X- 

^      7?      a^ 
2      3       4 

161 


4^8      16^32  ^^     ^  2»      ^      • 

16.  Prove  that  the  series  contained  in  Examples  11  and 
12  are  absolutely  convergent. 

17.  Prove  that  the  series 

-,    ,         ,  x(x  —  l^   o  ,  x(x—l)(x—2^  q  , 
1  4-  ^j/  +       2!       ^  ^     3! ^  "^  "* 

is  absolutely  convergent  when  |  y  |  <  1. 

213.  From  the  definition  of  195  the  following  propositions 
result: 

1.  An  absolute!  1/  convergent  settles  is  convergent  irrespective 
of  the  order  in  which  the  terms  are  taken. 

2.  The  sum  or  difference  of  two  absolutely  convergent  series 
is  an  absolutely  convergent  series^  and  the  two  series  may  there- 
fore be  added  or  subtracted  in  any  order, 

214.  Definition.  The  series  formed  by  taking  the  moduli  of 
the  terms  of  a  series  is  called  the  modular  series. 

If  the  modular  series  of  complex  terms  is  convergent,  the 
series  itself  is  also  convergent.     This  is  seen  by  using  the 

M 


162       ■  COLLEGE   ALGEBRA 

geometric  representation  for  the  addition  of  complex  num- 
bers (130),  in  connection  with  the  principle  that  the  modulus 
of  the  sum  of  two  complex  numbers  is  not  greater  than  the 
sum  of  their  moduli,  since  the  sum  of  two  sides  of  a  triangle 
is  not  less  than  the  third.  Thus  if  the  sum  of  the  moduli 
of  the  terms  of  a  series  is  finite  and  determinate,  so  much 
the  more  is  the  modulus  of  the  sum  of  the  series,  that  is,  the 

point  P  of  the  ratio  --— ,  tlie  sum  of  the  series,  is  a  determi- 

nate  point  lying  in  the  finite  region  of  the  plane. 

Theorem.  If  the  modular  series  of  real  terms  is  convergent^ 
the  complex  series  itself  is,  hy  theforegoiyig,  convergent^  and  is 
said  to  he  absolutely  convergent. 

215.  EXAMPLES 

Test  the  following  series  for  convergency  or  divergency: 
14  2n 


1. 


2'  7'      '3n  +  l' 


11 


7'  4'      '  2n2+a?i  +  2' 


oil  n^ 


8'  6'      '  »3+2w2+3,j  +  2' 


4       1     1     ...    -3 


2       '^2  +  2 


^         1  22  71^ 

5-     t:-, 


^    5  +  2^  ^2  +  w2  -h  1 


SEKIES  163 

12  n 


O  "  1       "  '  1 

2   11-2         (7i^+sy 


n—\ 


t 


^  V2  —  1  V«  —  1  „_-, 

VU  V/t^-2 


Test  the  series  whose  nth.  term  is: 


8. 


V?i  —  Vn  —  1 


^C^  —  1)  n-1 

10.       ^^ "^ X^    1 

(n  +  V){7i  -  2) (n  +  8) (n  -  4) 


CHAPTER   XII 
UNDETERMINED  COEFFICIENTS 

GENERAL   THEORY 

216.  Theorem.  If  the  infinite  series  a^  +  a^x  -\-  a^  +  a^:i^ 
4-  •••  -\-a^x^  -\-  •••,  Inhere  the  as  are  constants^  is  convergent  for 
a  value  of  x  other  than  zero^  and  is  equal  to  zero  for  all  values 
of  X  which  make  the  series  convergent^  then  the  coefficient  of 
every  power  of  x  is  zero. 

Proof.  If  the  series  is  convergent  for  any  particular 
value  of  X  not  zero,  it  is  also  convergent  for  a  value  of  x 
numerically  smaller,  198,  since  each  term  is  thus  made 
numerically  smaller.  Therefore  it  is  convergent  when  x 
approaches  zero  as  a  limit.  Hence  we  may  take  the  limits 
of  both  members  of  the  equation  a^-{-  a-^x  -^  a^x"^  -\-  a^oi^  -{-  ••• 
+  a^x^ -\-  •••  =0,  when  x=0.  In  order  to  take  these  limits 
we  write  the  equation  in  the  form 

a^  +  xS  =  0, 
where  S=  a-^-\-  a^x  +  a^x^  +  •  •  •? 

and  we  observe  that  x  -  S  \^  convergent  for  all  values  of  x 

which  make  the  original  series  convergent,  by  197.     Hence 

X  •  S  is  finite  and  determinate,  by  195 ;  if  a;  =  0,  aS'  is  equal 

xS 
to  flfj,  and  is  finite  and  determinate;  and  since  S= — ,    for 

X 

any  permitted  value  of  x^  S  is  always  finite  and  determinate 
for  all  such  values  of  x^  and  therefore  aS'  is  convergent,  by 
195.     Hence  x  •  S  approaches  zero  as  its  limit  when  x  ap- 

164 


UNDETERMINED   COEFFICIENTS  165 

preaches  zero.     Hence  La^-^-  Lx  -  S=()^hy  190,  2  ;  therefore 

a^  =  0,  and  since  a^  +  x  -  S  =0^  x  •  S  =0,  and  hence  as  a;^  0, 
aS'=0;  i.e.  a-^^  + a^x-^-a^x^ -{-'■■  =0.  By  a  repetition  of 
the  same  process  it  follows  that  <2j  =  0,  then  a^  =  0,  and  in 
general  a^  =  0,  for  every  positive  integral  value  of  n. 

217.  Theorem.  If  two  infinite  series  a^ -\- a-^x  +  a^^s^  + -- - 
-{■ajX^ -\-  '•'  and  bQ-{-  b^x  +  b^x"  +  •••  +5,^^+  •••  are  equal  to 
each  other  for  all  values  of  x  ivhich  make  both  series  convergent^ 
then  the  coefficieiits  of  the  corresponding  powers  of  x  are  equal. 

Proof.  We  have  given  a^ -\- a^x -{- a<f)^ -\-  •••  -{- a^x^ -\-  ••• 
=  5q  +  b-^x  +  b^x^  +  •  •  •  +  b^x^  4-  •  •  •  for  the  specified  values  of  x. 

Hence 

S— ^0+  («i  — ^i)2;  +  («2— ^2)^+  •••  +(^n— ^n)2:"+  •••  =0. 

Therefore,  by  the  preceding  theorem, 

S~^o  =  ^'  ^j  — ^1  =  0,  a^  —  b^=0,  ...,  a„  —  b^  =  0,  .••, 

or  aQ  =  bQ,  a^  =  b^,  a^=  b^,  ...,  a,^  =  br,,  ••••  (216) 

218.  Since  every  finite  series  may  be  regarded  as  an  in- 
finite series  in  which  all  the  terms  are  zero  except  a  finite 
number,  the  two  preceding  theorems  apply  to  all  series, 
both  finite  and  infinite,  of  the  same  form  which  satisfy  the 
given  conditions  with  respect  to  the  values  of  x.  Thus,  in 
the  preceding  theorem,  both  series  might  be  infinite  or  both 
finite,  or  one  finite  and  the  other  infinite.  If  there  are 
powers  of  x  in  one  series  which  are  not  found  in  the  other, 
the  coefficients  of  such  powers  must  be  equal  to  zero.     Thus 

^  aQ  H-  a-^x  +  a^x^  -\-  a^x^  =  ^0  +  ^1^  "'"  ^2^^ 

for  all  values  of  rr,  then 

a^  =  Jq,  a^  =  ^j,  ^2  =  ^2'  ^^^^  ^3  ~  ^- 


166 


COLLEGE  ALGEBRA 


DEVELOPMENT   OF  AN  ALGEBRAIC  FRACTION  INTO  A   SERIES 

219.  We  shall  now  use  the  preceding  theory  to  show  how  in 
certain  cases  an  algebraic  fraction  can  be  thrown  into  the  form 
of  an  infinite  series. 

EXAMPLES 


1.    Develop 


-  into  an  infinite  convergent  series. 


x^—  5  x-{-  6 
Assume,  if  possible,  that  the  fraction  is  equal  to  such  a 

series  ;  then  we  shall  have 

— —   "*"  —  =  a^  +  a^x  4-  ^2^^  +  •  •  •  +  '^,._2^^~^ 

2/    ^~*  O  u/  "J"  O  I  T—'\    I  V    I 

Clearing  of  fractions,  first  multiplying  by  6,  then  by  —  5  re, 
and  lastly  by  a:^,  we  have 

2  a;  +  1  =  6  «()  +  6  a^ 
—  5  an 


af-^ 


which  under  the  restrictions  of  the  assumption  furnishes 
two  series  satisfying  the  hypothesis  of  217.  Therefore, 
equating  the  coefficients  of  like  powers  of  x^  we  have 

1 

6 ' 

17 


+  6  «2 

^2  _|_    ...   _^  (3  ^^  _^ 

^r-2 

—  5  a-^ 

5a,_3 

+     % 

•••    +        «;.-4 

+  6  6«,,_i 

x'^-^  +  6  «,. 

—  5  <x,._2 

—  O  ^/.—-i 

+      «,-3 

+ 

a,_2 

6  6^Q  =  1,  .*.  fl^o  = 


6  ^j  —  5  ^Q  =  2,  .•.  6  «j  =  2  +  5  ^Q  and  t)^|  = 


79 


36 


6^2-5^1  +  S=  ^'  •'•  ^2  =  ^' 


6  a,.  —  5  a^_-^  +  ^;._2  =0,  if  r  >  1,  .*•  a,.  = 


5  a.,,_^  —  a,.. 
6 


UNDETERMINED   COEFFICIENTS  167 

x'^—Dx-^-b      o       db        216        129b 

values  of  x  as  make  the  series  convergent.     It  will  be  shown 
later  (230)  that  this  series  is  convergent  if  \x\<2. 

Note.  In  practice  we  may  write  the  first  term  by  inspection,  as  the 
ratio  of  the  terms  of  the  lowest  degree  in  the  numerator  and  the  denom- 
inator respectively. 

S  X  —  2 

2.    Develop  — — - — -  into  a  series. 

x^  —  4  a;^  -f-  2  ar 

Sx-2  1  Sx-2 


x'^  —  4:  X^  -{-  2  X^       X^      X^  —  4:  X  -\-2 

Sx  —  2 

Assume  — j =  «q  +  a-^x  -j-  a^x^  +  ^ga;^  H \-  a^x*'  -\ — 

As  in  the  preceding  problem,  we  find 

and,  in  general,  if  r>  1,  a''=  — -^^ ^^- 

Therefore,  to  five  terms, 

3  a;  —  2      _      ^      x      x^      o  x^      5  x^ 


x^-4x+2  2       2         1  1 

Dividing  both  members  by  x^^  we  have 

3a;-2  _l_J__l_i^_5^ 

a;*  —  1  a:^  -f-  2  a;^  x^      2x      2       4  4 

for  such  values  of  x  as  make  the  series  convergent.  It  can 
be  shown  that  the  series  is  convergent  for  all  values  of  x 
numerically  between  0  and  0.5858+. 


168 


COLLEGE   ALGEBRA 


3.    Develop  -- — - — - — - — -  into  a  series. 

By  inspection  we  see  that  the  first  term  is  -  x~^.     Hence, 

3 

since  the  series  must  be  in  ascending  powers  of  x  beginning 
with  x~^,  we  assume 


X  t:  X  -f-   J-  -t       o  j^  2     I  —1 

a^-'Zx'^  +  Sa^      ^  ^  ^  ^ 


4-  a^^x^  4- 


Clearing  of  fractions, 

2 

~5 


X  -\-  S  a_j 
-2a 


-2 


+ 


rz:^  +  3  a. 


_  9 


a. 


+  «_r 


a:^ 


+  3ai 
-2a, 
+    a_. 


a:*  H h  3  a. 


z  a 


r-l 


+     a 


r-2 


X'^^-^- 


10 


14 


Therefore    a_^=  - —- ;  a_i=-  — ;  a()  = 


81 


;  <^i 


243' 


and,  in  general. 


^^^^a,_i^   ^^-2^^>_1. 


Hence      ^^-4:r  +  l    ^1         10        14        2       46^ 
a^-2a^  +  Sa^      S  x^      d  x^      27  a;      81       243 

for  such  values  of  x  as  make  the  series  convergent. 


UNDETERMINED   COEFFICIENTS  169 


220.  EXAMPLES 

Develop  to  four  terms  the  following  fractions  into  series 
in  ascending  powers  oi  x: 

^  l-\-Sx ^        2-Sx  +  x'^ 


^     5x^  +  2  ^  ^+8 


3^—4:  X^—2x-\-  4: 

-3a;2+7a;  .      a-{-bx 

o. •  b.      . 

X  -{-  4:0^  c  +  dx 


BINOMIAL   THEOREM,    ANY   REAL   EXPONENT 

221.  We  shall  now  complete  the  proof  of  the  binomial 
theorem  as  we  promised  in  171.  We  consider  two  cases, 
according  as  7i  is  commensurable  or  incommensurable. 

Firsts  when  n  is  any  commensurable  number  whatever, 
positive  or  negative,  integral  or  fractional. 

Assume  (1  +  re)"  =  ^^  +  a-^x  +  a^  4-  a^x^  +  •  •  •  +  drX"^  +  •  •  •    (1) 

Then      (1  +  yj'  =  ^o  +  HV  +  Hlf'  +  HV^  ^ ^  ^rV''  H • 

Subtracting,  and  dividing  by  (1  +  a?)  —  (1  +  ^)  =  a;  —  y,  we 
have 

(l4.^)n_(i^yy^^  fx^-jy\         fx]^-f\         fo^-f 
(l  +  2;)-(l+^)  \x-yj       \x-yj       \x-y. 

\x-y  J 

Taking  the  values  of  both  members  when  y  —  x^  and  there- 
fore also  1  +  ?/  =  1  +  ^,  we  have,  by  192, 

n(\  +  xy-^  =  a^  +  2  «2^  +  3 a^x^  H (-  ra^"^  H • 


170  COLLEGE   ALGEBRA 

Multiplying  both  members  of  this  equation  by  1  +  re, 

w(l  +  xy  =  a^  +  (^1  +  2  a^^x  +  (2  ^2  +  3  a^')x'^  +  •  •  • 

-\.[(r-l}a,_^  +  ra,']x'-'^ -{-■•'.    (2) 
But 

n(l-{-  x^  =  na^  +  na^x  +  na^x'^  +   •  ■  +  ?^(2^_Ja;^~l  +  •  •  •     (3) 

by  equation  (1). 

Since  things  equal  to  the  same  thing  are  equal  to  each 
other,  the  two  infinite  series  forming  the  right  members  of 
equations  (2)  and  (3)  are  equal  to  each  other  for  all  values 
of  X  which  make  both  series  convergent.  Therefore,  by 
217,  we  may  equate  the  coefficients  as  follows : 

2  ^2  +  ^1  =  ^^^1' 

3  C?3  +  2  ^2  =  ^^<^*2' 


ra^  +  (r  —  1  )c/y_i  =  na^_i. 

But  from  equation  (1),  by  ]3utting  a;  =  0,  we  find  aQ  =  l. 

mu       £                                     nCn  —  V)             n(n—V)(n —  2?) 
Therefore  a^  =  n,  a^  =  -^ — — ^,  a^  =  ^ —      -;^   ^, 

1-2  1  • 2 • 3 

and,  in  general,  a,.  = •  «,._-, . 

r 

Changing  r  into  r  —  1  in  this, 


^^  -  r  +  2 
''-1=     r-l 

•  ^r-2? 

UNDETERMINED   COEFFICIENTS 


171 


Multiplying   corresponding  members  of   these   last   r  +  1 
equations,  and  canceling  the  common  factor  a^a-^a^a^-'-ar.-^^ 

we  have         .        ^^^        r,.       .  ^x/> 

__  n(n—  l)(n  —  z)  ••  •  (^z  —  r  +  1 )  _  ni 

~  1.2.3-^  ~  W. 


a,. 


Hence  our  development  is 

(i+.)»=(«)+(»).+g)..+  ...+(»y+....  (4) 


It  remains  to  consider  the  convergence  of  this  develop 

ment.  ^  n       /         s  -. 

n\     (    n    \      n  —  r  +  \ 

.r-1 


Since 


Wy+i  _  ?t  —  r  +  1 


X. 


u. 


Therefore 


(172) 
(cf.  183) 


'•+1 


Ur 


Hence  the  development  is  convergent  if  la:|<l. 


(206) 


If 


a:  =  - ,     then 
a 


<1;    that  is,     |J|  <  la 


Substituting  -  for  x  in  (4),  we  have 
a 

Multiplying  both  members  of  this  equation  by  a"^, 


which  is  convergent  provided  |5|  <  la|. 


(5) 
(cf.  171,  3) 


172  COLLEGE   ALGEBRA 

Second,  when  n  is  any  incommensurable  number. 
Let  m  be  a  commensurable  number  which  approaches  n  as 
its  limit.     Now,  by  (5),  if  |5|<  |a|, 

(a  +  hy^  =  h^  a^'^  +  ("f]  a^-^h  +  C^\  a'^-%'^  +  •  •  • 

+  h\a^-'h'  +  '",     (6) 
But  L  (a-\-  by  =  (a  +  ^)^ 


m=n\  r 


L  i''%^-'b'  =  r]a^~'b^. 


Denote  the  series  in  (6)  by  aS'^  and  the  series 

^y«  +  ('!)^"-i^  +  •••  4-  C^)^^-'-^'-  + 

by  S.     Then  we  have 

L  SJ  =  iS'„  by  190,  2. 

Hence  i  (L  S,')  =L  S,  =  S; 


r=co  m—n 


that  is,  we  have  proved  that  S'  approaches  S  as  its  limit  as 
m  approaches  n ;  and  also  (a  +  5)"*  approaches  (a  +  ^)"  as 
its  limit ;  but  (a  +  5)""  =  /S"  always,  therefore 

{a  +  by=-S,  (190,1) 

(a  +  J)-  =  {^\  a-  +  r^^)  a^-^b  4-  Q  a"-2J2  +  . . . 


or 


Another  proof  is  given  in  378. 


+  f^U''"^^''+  •-. 


UNDETERMINED   COEFFICIENTS  173 

222.  EXAMPLES 

Expand  to  five  terms : 

1.  (x  +  y)^.  7.    (a^-^^)?. 

2.  (2x  +  2>y~)\  8-    (2  +  3x2)-2. 

3.  (3a;2+2?/3)i  9.     Ux^-X^- 


10. 

5.      (2-0.2)1.  Va2         2/2 

.    _gs|  11.  (2  a;- 3^2)-!. 

^'      V     "^J    *  12.  (2:2-2^3)-|. 

Develop  into  a  series  to  a  term  containing  as  high  as  the 
fourth  power  of  x : 

13.     VI  +  3  a:.  15.  Vl  +  2  a:  +  3  a^. 


14.    V2  +  3  2;2.  16.    V2-5a;2. 

Develop  to  five  terms  : 


17.    V2  a  +  3  6  in  ascending  powers  of  h. 


18.  ■\/2  a  +  3  6  in  ascending  powers  of  a. 

19.  Find  the  coefficient  of  the  7th  term  in  V3  —  2  x2. 


20.    Find  the  coefficient  of  x^^  in  V3  —  2x^. 

DECOMPOSITION  OF   FRACTIONS   INTO   PARTIAL   FRACTIONS 

223.  In  certain  mathematical  investigations,  as  in  the 
integral  calculus,  it  is  often  required  to  express  an  algebraic 
fraction  as  the  sum  of  several  others  whose  denominators 
are  of  a  simpler  form  than  that  of  the  given  fraction.     Thus 


174  COLLEGE   ALGEBRA 

it  may  be  required  to    express  — — as   the    sum    of 

x^  —  0  x-\-  6 

such  fractions.  Since  the  denominator  is  composed  of  the 
factors  (re  —  2)  and  (:r  —  3),  we  take  these  factors  as  the 
denominators  of  the  required  fractions,  and  assume  if  pos- 
sible for  all  values  of  x 

2a;+l      ^     A  B 


x^—  5  X  -\-6      X  —  2      X  —  S 

where  A  and  B  are  undetermined  constants. 
Clearing  of  fractions,  we  have 

2  a;  +  1  =  A(x  -  3)  +  Bi^x  -  2).  (1) 

Equating  coefficients  of  like  powers  of  x,  (217) 

-3^-2^  =  1, 
A  +  B=2. 

Solving  for  A  and  j5,  we  have  J.  =  —  5,  ^  =  7,  for  all  values 

of  X. 

u                               2a;  +  l             7            5 
Hence  — — — ^ ~  = -• 

X^  —  D  X  +  b         X  —  €)        X  —  I 

Thus  our  assumption  is  justified ;  for  it  requires  us  to 
find  the  values  of  two  unknown  quantities  when  two  inde- 
pendent equations  of  the  first  degree  containing  them  are 
given.  This  always  gives  a  determinate  solution  for  each 
unknown. 

We  might  have  obtained  the  values  of  A  and  B  by 
another  method.  Since  equation  (1)  is  true  for  all  values 
of  X.  it  is  true  when  2:  =  2  and  when  a:  =  3. 


UNDETERMINED   COEFFICIENTS  175 

When  a;  =  2,  (1)  becomes       (2  x  +  1)^=2  =  ^(^  —  3)^2' 
whence  A  =  l ^t^)      =—5. 


X  —  3   J  x=2 

Similarly,  B  =  ('^l^±1)      =  T. 


X  —  '2   /x=3 

Observing  the  form  of  A  and  B  which  we  have  last  ob- 
tained, we  see  that  A  is  obtained  by  putting  x=  2  in  a 
fraction  which  is  derived  from  the  original  fraction  by  neg- 
lecting the  factor  x  —  2  in  the  denominator.  Likewise  for  B. 
And  in  general  it  may  be  shown  that  the  same  law  for  find- 
ing the  constant  numerators  always  holds  when  the  denom- 
inator is  a  product  of  different  linear  factors.  The  value 
of  X  to  be  used  to  find  a  required  numerator  is  that  which 
makes  its  denominator  vanish. 

224.  Four  forms  of  fractions  are  treated  according  to  the 
nature  of  tlie  fa,ctors  of  their  denominators.  Firsts  those  ivhose 
denominators  consist  of  different  linear  factors ;  second^  those 
whose  denominators  are  poivers  of  linear  factors  ;  thirds  those 
whose  denominators  consist  of  irreducible  quadratic  factors ; 
fourth^  those  whose  denominators  consist  of  combinations  of  the 
factors  mentioned  in  the  first  three  cases. 

225.  Since  every  rational  fraction  whose  numerator  is  not 
of  lower  degree  than  its  denominator  can  be  reduced  to  the 
sum  of  an  integral  expression  and  a  fraction  whose  numera- 
tor is  of  lower  degree  than  its  denominator,  the  decomposi- 
tion of  only  this  type  of  fraction  is  treated. 

226.  The  forms  into  which  the  types  of  fraction  men- 
tioned in  the  four  cases  of  224  are  to  be  decomposed  into 
partial  fractions  are  as  follows : 

1.  A  fraction  of  the  first  type  is  decomposed  into  a  sum 
of  fractions  whose  denominators  are  the  several  linear  factors 


176  COLLEGE   ALGEBRA 

of  the  original  denominator  and  whose  numerators  are  un- 
determined constants.     Thus 

-r 


x^-Qx^-^llx-6      (a;- l)(a;-2)(2;-8)      x-1      x-2 

+  — ^• 
x  —  Z 

2.  When  the  denominator  of  the  fraction  to  be  decom- 
posed is  a  power  of  a  linear  factor,  the  denominators  of  the 
partial  fractions  are  the  different  powers  from  the  first  up 
to  the  power  occurring  in  the  original  denominator,  and  the 
numerators  are  undetermined  constants.     Thus 

x^-\.x-^r\  A       ^       B      ^     0 


{x-\f       {x-Vf      {x-Xf     x-1 

The  reason  for  this  statement  is  evident  without  a  formal 
proof.     The  left-hand  member  may,  by  85,  be  written  in  the 

degree  one  lower  in  x  than  the  original  numerator.     Like- 
.  p       _0(x-l)-\-B_       B  0 

^^^^  (x-iy~    (x-iy    ''{x-iy   x-i 

3.  In  this  case  the  denominators  of  the  partial  fractions 
are  the  several  quadratic  factors  of  the  original ;  the  numer- 
ators are  binomial  expressions  of  the  first  degree  in  x,  each 
containing  two  undetermined  constants.     Thus 

x^-\-x^-{-x-hl       _Ax  +  B        Cx-\-I> 


(a;2  +  3)(rz^2_^a^+l)        x^-^S        x^-^x-\-l 

An  expression  of  the  form  x"^  -^  ax  +  h  is  equivalent  to 
(a;  +  «)  (a: -F  yS)  where  a  and  13  in  this  case  will  be  surds  or 
imaginaries,  and  conjugates  of  each  other.     Hence  the  frac- 


UNDETERMINED   COEFFICIENTS  177 

tion  whose  denominator  is  x^  +  ax  +  b  is   the   sum   of  two 

A'          B' 
fractions  of  the  form 1 -,  and  hence  must  itself 

X  -\-  a      x-\-  p 

Ax  -\-  B 
be  of  the  form  — — -•     For  a  full  discussion  of  these 

x^  -\-  ax  -\-o 
special  types  see  228. 

4.  In  this  case  the  original  fraction  is  decomposed  into 
the  sum  of  all  the  partial  fractions  to  which  each  factor  in 
its  denominator  gives  rise,  as  stated  in  the  three  preceding 
cases.     Thus 

x^  +  ^x-\-l  _    ^1  A  A 


(a;- l)3(a;- 2)2(0^2^4)      a;-!      (2: -1)2      (x-iy 

B,  B,  C,x+C, 

x-2.      Qx-2y        2:2  +  4 

227.    We  complete  the  solution  of  the  first  three  problems 
of  226. 

1.    By  the  second  method  in  223, 


\{x-2)(x-n)J,^, 


—  o 


(-l)(-2) 


B  =  l    2a^-^+3    ^      =_^  =  _9, 


(^_l)(^_3)y^^^     1(_1) 

^^f    2x^-x  +  S    \       ^      18      _g 
\(x-lXx~2)J,^,     (2)(1) 

Hence     „    2^-^  +  3        ^^ 9_^       9 


afi-63?  +  nx-6      (:»-l)      i^--}      (^-3) 

2.    The  second  example   may  be  solved  as   in   the   first 
method  of  223  by  clearing  of  fractions  and  equating  coeffl- 

N 


178 


COLLEGE   ALGEBRA 


cients.  We  may  also  use  a  modification  of  the  second 
method.     Thus  after  clearing  of  fractions  we  have 

x^-hx-{-l  =  A  +  B(x-l')-\-  C(x-iy. 

When  X  =  1  this  becomes  A  =  (a;^  -i-  x  i-  l)j;=i  =  3. 

Thus  A  is  obtained  by  putting  x  =  l  in  a  fraction  which  is 
derived  from  the  original  fraction  by  neglecting  (a;  —  1)^  in 
its  denominator.     Thus 


x"^  -{-  x  +  1  _ 


+ 


B 


-f- 


C 


{x-if     {x-\y    (x-if    x-i 

Transposing,  combining,  and  dividing  out  the  factor  x  —  1, 


x+2 


B 


+ 


0 


(ix  -  1)2        (x  -  1)2    ■    ^^  _  1 

Proceeding  as  before,         B  =  (x-{-  2)^.^i  =  3. 

Putting  in  this  value  of  B,  and  transposing  as  before, 

1     _     (7 


Therefore, 


X  —\      X  —  1 

x^  +  x+1  3 


whence  0=1. 


+ 


T,+ 


3. 


(2; -1)3     (x-iy    (x-iy    x-i 

a:3  +  2^2  +  a:  +  1       _  Ax -\-  B  ,      Cx  +  D 


+ 


(a:2+3)(a:2  +  a:+l)        x^  +  ^        x^  +  x-^1 


Clearing  of  fractions, 

x^  +  x'^-\-x-\-l  =  A 
+  0 


a:3  +  ^ 

x^+    A 

+  B 

+    B 

+  i> 

+  3(7 

X 


+     B 

+  3i). 


UNDETERMIXED   COEFFICIENTS  179 

whence 


Subtracting  (2)  from  (1), 

Subtracting  (3)  from  (2), 

Adding  (4)  and  (5), 

Subtracting  2  x  (6)  from  (7), 

And  by  substituting,  i^  =  f ,  B  =  —  ^,  A=^. 

Therefore 

x^  -{-  x^  +  X  -\- 1       _    G  X  —  2  X  +  S 


A         \     G           =1, 

0) 

A^B           +     Z»=l, 

(2) 

^  +  fi+3C            =1, 

(3) 

B            +8i)=l. 

(■t) 

-  ^  +     G  -     i>  =  0. 

(5) 

-  8  C  +     2>  =  0. 

(6) 

C+2i)  =  l. 

(7) 

),                         G  =  f 

(2:2  +  3)(2:2  +  a:  +  l)       7(^:2  +  3)       7(^:2  +  2:4-1) 

228.  T/te  decompositions  stated  in  226  <?ari  he  shown  to  he 
correct  hy  oh  serving  that  the  numher  of  unknown  constants 
introduced  is  the  same  in  each  case  as  the  numher  of  equations 
arising  hy  equating  coefficients  for  the  solution  for  the  unkriown 
constants^  and  that  the  solutionis  are  in  each  case  unique. 

If  the  denominator  of  the  original  fraction  is  of  the  degree 
?^,  each  of  the  various  cases  will  give  rise  to  n  undetermined 
constants.  When  the  partial  fractions  have  been  reduced 
to  a  common  denominator  and  added,  the  numerator  will  be 
of  the  (>i—  l)th  degree,  thus  giving  rise  to  n  equations  for 
the  determination  of  the  n  constants.  But  the  agreement  of 
the  number  of  constants  and  the  number  of  equations  is  not 
sufficient  to  show  that  the  solutions  will   be  unique.     For 

this  purpose  we  consider  the   fraction   ^ — -^  where /(a:)  is 

of  degree  w,  and  <^(x^  of  degree  (n  —  1).     Let  fi^x^  contain 
r   different  linear    factors    (x— «j),    (x—a^^    •••,    (a:  — a^); 


180 


COLLEGE   ALGEBRA 


s  sets  of  factors  (x  —  P^\   (^x— P^\   •••,   (x— ^sYs],    and 
t  sets  of  factors  (^^  +  7i^),  (^'^  +  72^)?  ••>  C^^  +  T^^.* 
Then,  by  226,  we  ought  to  have 


+  •••  + 


^, 


+ 


B, 


B. 


x-ar      (x-p^y^      (a;-/3i)'i-i 


+ 


X—  /3i 


+ 

+ 


+ 


^1 


+ 


(:.-yS,y.    (x-^,ys-^ 


+ 


X, 


^- A 


H »  H ^^ ^ 


x^  +  Ti''^ 
+  •••  + 


a;2  +  72^ 
^  +  7<^ 


Here  we  have  r-{-r^-{-r^+  •  •  •  +  r^  4-  2  ^  =  7^  constants.  The 
second  method  of  223  and  the  method  of  Example  2  of  227 
show  that  the  ^'s,  -6's,  •••,  Z's  can  be  determined  uniquely. 
As  to  the   (7's  and  D's,  instead  of  the  fractions  of  the  form 

— '^ — ,  we  might  have  taken  fractions  of  the  form 
Q^  +  7^ 


R, 


s. 


-.+ 


XLr 


rr  +  7l^      x  —  ^^%      x -\-  y^i      x  —  y^z 


-.+ 


^o 


-,+  •••  + 


Bt 


^t 


a;  +  7i^      x—  7^1 


by  the  first  case,  where  the  M's  and  S's  would  now  be  among 
the  A's  and  therefore  could  be  uniquely  determined. 

Each  pair  like  M^  and  aS'j  would  be  complex  conjugates,  for 


''•=[/g)^^^^^^'^' 


,    S^  = 


Vi* 


Mx) 

L/(^) 


(x  -  7i0 


=Vi» 


*  Since  any  quadratic  expression   ax"^  -\-  bx  -{-  c  can  be  thrown  into  the 

form   f  Vax  +  — ^  )    +  c  — — ,  the  treatment  of  factors  of  this  type  is  con- 

V  2Va/  4a 

tained  in  that  here  given. 


UNDETERMINED   COEFFICIENTS  181 

Put  f(x)  =  [^|r  (x)]  (2:  +  7i0  (x  -  7^0 . 

Then     E  = ^("^lO  S  =  ^CtO  . 

'      2[^/.(-7i0](-7i0'       '      2[^/.(7i0](7i0' 

that  is,  i^i  and  ^j  come  from  each  other  by  changing  i  into 
—  i  in  the  function  which  expresses  either.  Hence,  by  118, 
H^  and  jS-^  are  conjugates  and  hence  of  the  form  a  +  bi  and 

a  —  ^^. 

TT                  M.       ,       )S.           a-\-hi    ,    a  — hi       2(ax  -{-  hy.^ 
Hence      ^  H ^  =  — ^ .  H .  =  -^— fLZ . 

a;  -f  7jZ      2:  —  7J^      2:  +  7i^      ^  —  7i^  ^  +  7i 


Whence  Cj  =  2  «,     i)i  =  2  by^ 

In  like  manner  similar  conclusions   c 
regard  to  any  other  proposed  type,  including 


In  like  manner  similar  conclusions   can  be   obtained    in 

Ux-{-F 


229.  EXAMPLES 

Decompose  into  partial  fractions  : 


^  4:x^-59x-29  5r^-{-6x^+5x 

Qx-{-2){x-b)QSx  +  iy      '    {x^-l)(x-^l) 

2.        .     ^  +  ^ 8.  ^  +  ^ 


3 


{x^  +  1)  (a;  +  2)  15  x-^  2x^-0^ 

7rr3-15a;4-12  22:-l 

{x-\-l)Xx-iy'  ^' 

a;2  —  7  a;  +  12 
2a:4_3^^7 

a;4-l         '  '    (2:2+l)(a;2  +  2'H-l) 

*    (2a;-l)(a:  +  2)2  *    x^ -{- x^  -  2 


(5x 

+  7)c3:.-2)(4- 
a:2+l 

-^) 

(X- 

1 

182  COLLEGE  ALGEBRA 


13.  _^izLi_.  17.      6x^-\-x-l 


x^  +  x^+1  (2^2  +  IX^  -  2)(2:  +  3) 

14.  J,„   ...         18.       ^+1 


15. 


{x  -  l)2(a;2  +  1)2  (:c2  _^  i)2(^  _  i^^ 

^6  _  1  2:4-1 


x^-x-^1  x^  + 2x^-2 

^*  (^^-iy(^x-2){x^  +  l)  '    x^  +  x^-2 


GENERAL  TERM  IN  THE  DEVELOPMENT  OF  AN  ALGEBRAIC 

FRACTION 

2  2;  +  1 
230.    In  219  we  have  developed       "" into  a  series, 

X^  —  0  X  -^  D 

and  have  given  the  law  of  relation  connecting  any  three 
consecutive  coefficients.  We  shall  now  obtain  a  single 
explicit  expression  for  any  coefficient  independently  of  the 
other  coefficients. 

By  223, 

22:+l      ^     7 5___5 7_ 

a;2— 5a?  +  6      x  —  o      x  —  2      2  —  x      S  —  x 
=  5(2-0:)-!- 7(3 -2:)-i, 

which  we  can  expand  by  the  binomial  theorem  into  a  con- 
vergent series  provided  |rr|  <  2. 
We  have 

^^  +  ^      =5(2-1+2-22;+  ...  +2-i-^x^^+  ...) 

X^  —  b  X  -\-  6 

_  7(3-1+ 3-2 ^^_  ...  _^3-i-.^.r^  ...). 

Hence  the  coefficient  of  x''  or  a,. 

5  7 


=  5.2-1-^-7.3-1-^  = 


2^+1      3''+i 
for  any  value  of  r. 


UNDETERMINED   COEFFICIENTS  188 

231.  EXAMPLES 

1.    Use  the  formula  of  230  to  obtain  the  first  five  terms 

of  the  development  of  — -^ — —  into  a  series. 

x^  —  o  X  -\-  0 

Find   the  general  term   in,  and  obtain  five  terms  of   the 

development  of : 

^  X  -{-1  S  X 


x^-Sx  +  2  x^-2x-{-l 

SUMMATION   OF   INTEGRAL   SERIES 

232.  We  shall  use  the  method  of  undetermined  coefficients 
to  obtain  the  sum  of  7i  terms  of  a  series  whose  nth  term  is  of 
the  type  ,  ^_, 

Thus  to   find  the   sum   of   the   first  n  terms  of   the   series 

12  _j_  22  _|_  32  _|_  ...  _|_  ^^2^  ^yg  assume 

12  +  92  +  ...  +  ?i2  =  ^  +  Bn  +  (7n2  +  Dti^  +  IJn^  +  •••      (1) 

as  true  for  all  values  of  n.  It  is  therefore  true  for  the  next 
higher  value  of  7i. 

Hence     I2  +  22  +  32  +  •  •  •  +  «2  _^  ^^^  ^iy  =  j_-\-  B(n  +  1) 

+  C(n  +  1)2  +  I)(7i  +  1)3  +  ^(n  +iy+  •••.  (2) 

Subtracting  (1)  from  (2), 

(^i  +  1)2  =  ^  _^  (7(2  n  +  1)  +  i>(3  ?i2  +  .3  n  +  1) 

+  ^(4>i3  +  6m2  +  4>/  +  1)+  •••. 

Since  this  is  an  identity  in  which  both  members  are  finite 
series,  the  degree  of  both  must  be  the  same,  that  is,  the  second 
degree,  whence  it  appears  that  all  coefficients  like  U,  F^  ••• 


184  COLLEGE   ALGEBRA 

must  be  equal  to  zero.      Equating  the  coefficients  of  the 
corresponding  powers  of  n  in  the  remaining  terms,  we  have 

B-{-    C+    D  =  l, 

2  (7  +  3  D  =  2, 

3i)  =  l, 

whence  -^  =  3'     ^  =  2 '     ■^  —  \' 

Substituting,    I2  +  22  +  32  +  ...  +  /i2  =  ^  +  l  ^  +  1  ^2  +  1  ^3, 

Now  putting  n  =  1,  we  have  1  =  J.  +  1 ;  therefore  ^  =  0. 
Reducing  to  a  common  denominator  and  factoring, 

12  +  22+32+  ...  +ri2  =  ^(2nH-l)(n  +  l). 

b 

Similarly  the  sum  of  n  terms  of  any  other  series  of  similar 
type  can  be  obtained.  It  is  to  be  noted  that  the  series  to 
be  assumed  as  the  sum  of  n  terms  is  to  be  of  a  degree  one 
higher  than  that  of  the  nth.  term  of  the  given  series. 

Thus 

1.2.3  +  2.3.4+  ...  +m(72  +  1)(^  +  2) 

=  A  +  Bn-\-  Cn^  +  Dn^  +  EnK 

233.  EXAMPLES 

Find  the  sum  of  n  terms  of : 

1.  I  .2  +  2.3  +  3  .4+  .... 

2.  1.2.3  +  2.3.4  +  3.4.5+  .... 

3.  13  +  23+33+....  5.    12  +  32  +  52+.... 

4.  14  +  24  +  34+....  6.    1.22+2.42+3.62+.... 

7.  1  .2.4  +  2.3.5  +  3.4.6+  .... 

8.  The  series  whose  nth.  term  is  2  n^  +  3  /i  +  1. 


CHAPTER   XIII 

CONTINUED  FRACTIONS 

234.    Definition.    An  expression  of  the  form 

h 


a  -f- 


d 


e^      ^ 


9+"- 

which  is  for  convenience  tvritten  in  the  form 

h      d     f 

a  +  -      -      -••• 
^+  e  +  g 

is  called  a  continued  fraction. 

Continued  fractions  fall  into  two  classes,  those  which  te?'- 
minate  and  those  which  do  not.  The  latter  class  is  called 
infinite  continued  fractions.  Of  the  first  class  we  shall  con- 
sider here  only  those  which  are  in  the  form 

111  1. 

^      a^-\-  a^-\-  a^+       +  a„ 

A  more  convenient  notation  for  this  is   (a^,  a^^  ••-,«„).     If 
-  denotes  the  value  of  the  continued  fraction,  we  have 

0 

.  .  V 

235.  Theorem.  It  is  possible  to  express  any  fraction  — >  P 
and  q  being  integers^  as  a  continued  fraction. 

185 


186  COLLEGE    ALGEBRA 


7)  V  1 

For,  ^  =  a.  -\-  -^  =  a^-{-  -  ' 

q         ^       q  q 


Continuing  this,  we  have 

^71 

1 

Li 

1 

1 

^2  + 

a, 

1 

«2,  •••,«„). 

It  is  to  be  observed  that  the  process  here  used  is  the  same 
as  that  employed  for  finding  the  highest  common  factor  of 
the  two  numbers  p  and  q  and  therefore  will  terminate, 

The  student  must  not  fail  to  observe  that  when  p  <q,  a^  = 
0,  and  rj  =  p. 

EXAMPLES 

1.    Reduce  |J|  to  a  continued  fraction. 
The  work  is  as  follows: 


629)271(0 
000 

128 
117 

271)529(1 
271 

11)13(1 
11 

258)271(1 

258 

2)11(5 
10 

13)258(19 
13 

128 

1)2(2 
2 
0 

Hence  aj  =  0,  a^=  1,  a^  =  1,  ^^  =  19,  a^  =  1,  a^  =  5,  ^7  =  2, 
and  we  have  ||i  =  (0,  1,  1,  19,  1,  5,  2). 


CONTINUED   FRACTIONS  187 

2.    Reduce  ^l  to  a  continued  fraction. 

Proceeding  as  in  Example  1,  Ave  find  J||  =  (2,  4,  3,  1,  18). 

236.    Let  ^  be  denoted  by  ll, 
1  9i 


l^a,^,+  l         g 


^1  + T-  =    '  '  ^     ^^ ^  by  ^-^\  etc., 

where  in  each  case  the  ^'s  denote  the  numerators,  and  the  ^'s 
the  denominators,  then  we  define  li,  1^,  li,  etc.,  as  the/rs^, 

second^  thirds  etc.,  convergents  of  the  continued  fraction. 

237.    The  law  of  formation  of  these  convergents  may  be 
found  inductively  thus : 

p^      a^a^a^  +  <^3  +  «i      ^^zPi'^  Pi 

P4^^_4PS±P2      g^^ 

Let  m  be  a  value  for  which  this  law  has  been  observed  to  hold. 

rpi  Pm ^mPm  —  l    >    Pm—2 

9m  ^m9m-\  ~r  9m- 2 

Now  -^^  is  obtained  from  —  by  substituting  a„^  -\ in 

9m+\  (j[m  ^m+\ 

place  of  a^.       /  -j    n 

I  a^  -\  j  ]^m-\  +  Pm-2 

ence   =  y :j — r 

^m  +  l\^mPm-l    i    pm-l)  +  Pm-\  ^m+\Pm+  Pm-\ 

^m+l(^^m9m-l  +  9m-2)  +  9m-l  ^m+\9m    '    9m-\ 


188  COLLEGE   ALGEBRA 

Therefore  the  law  holds  for  -^^,  and  since  it  holds  for  m 

=  3,  it  holds  for  m  =  4,  then  5,  and  so  on,  and  thus  it  holds 
for  all  applicable  integral  values  of  m  greater  than  two. 

Hence  the  theorem  Pni  =  ^7nPm-i+ Pm-2^  a-nd  qm  =  cCm<im-i 
H-^m-2  when  m>2. 

238.  EXAMPLES 

1.  Find  the  successive  convergents  to  ||^. 

We  have  found  in  Example  1,  235,  that  a^  =  0,  ^2  =  1,  a^  =  1, 
a^  =  19,  a^  =  1,  ag  =  5,  a^  =  2.  Hence  the  convergents  are 
^,  h  h  ti  It'  iff'  fl4-  The  whole  work  is  checked  by 
finding  the  last  convergent  equal  to  the  original  fraction. 

2.  Find  the  successive  convergents  to  J^|. 

3.  Find  the  successive  convergents  to  ||^. 

4.  Convert  -^Jj-  into  a  continued  fraction,  and  check  the 
work. 

5.  Convert  ^^j^-  into  a  continued  fraction,  and  prove  the 
correctness  of  the  work. 

239.  Definition.  A  recurring  continued  fraction  is  a  non- 
terminatiyig  continued  fraction  in  which  from  a  certain  point  on 
a  number  of  quotients  periodically  recur.     Thus 

54-1      11111 

"^4  +  3  +  2  +  7  +  2  +  7+*" 

A  1111111 

3  +  4  +  3  +  4+3  +  4  +  3  +  '" 

are  examples  of  recurring  continued  fractions.     In  the  second 
the  quotients  recur  from  the  beginning. 

240.  Theorem.  Every  recurring  continued  fraction  is  a  root 
of  a  quadratic  equation. 


CONTINUED   FRACTIONS  189 

Let  the  fraction  be 
«  +1       ...       i      1      1       ...      1      1      1  1       .. 

or,  as  it  may  be  written,   (a^,  a^,  •••,  a,.,  5j,  h^,  •••,  6^). 

Let  X  denote  the  entire  continued  fraction,  and  y  denote  the 
recurring  portion, 

6+111  1 

1  hen  x=  a^-] —      —      •••      —      — . 

^2  4-  «3  +       +a^-\-y 

Then,  by  237,       x=^f±^. 

p 

where  — ^  is  the  rth  convergent  of  x. 

Hence  y=&L±Ilzl, 

and  ^,?/2  +  (g,_^  -Ps^y- Ps- 1  =  0, 

where  ^^  is  the  sth  convergent  of  y. 

Substituting  the  value  of  y  in  terms  of  x  as  obtained  above 
in  this  equation,  and  clearing  of  fractions, 

qs{Pr-l  -  Qr-l^y  +  fe-1  -PsXPr-1  "  Q.-l^XQr^  "  ^r) 


190  COLLEGE   ALGEBRA 

or        [^.^r-l^  +  iPs  -  qs-l)  QrQr-1  -  Ps-lQ>^]^'^  -  [2  qs^r-lQr-l 
+  iPs-qs-l){Pr-lQr+PrQr-l)-^Ps-lPrQr']x 
+  qsPr-l    +  {.Ps  -  qs-l)PrPr-l  "  Ps-lP,^  =  ^' 

Example.    Find  the  quadratic  equation  of  which 

1111 

3  +  4  +  3  +  4+'" 
or  (0,  3,  4)  is  a  root. 

Let  -  equal  the  value  of  the  fraction. 

y 

rpi  1111 

1  hen  -  =  -      —      - . 

y      3+4+^ 

Here  iLi  =  _;       -Li  = — . 

qi      3         ^2      13 

Whence  -  =  — ^—^ — ,  or  if  x=  -^ 

y      13j/+3  y 

3  0^2+12  2: -4=0. 

c  1   •      +!.•           +•                      ±4V3-6 
bolvmg  this  equation,  x  = . 

3 
Since  the  continued  fraction  is  manifestly  positive,  its  value 
.    4V8-6 

241.  EXAMPLES 

Find  the  quadratic  equations  of  which  the  following  are 
roots,  and  exj)ress  these  roots  in  surd  form: 

1.  (0,  2,  5).  3.   (1,  2,  3). 

2.  (0,  3,  2).  4.   (3,  5,  2). 

Suggestion.     Put  x  —  S  =  y. 

5.   (2,  5,  7).  6.   (1,  2,  3,  4,  5). 


CONTINUED   FRACTIONS  191 

242.  It  may  be  proved  tliat  every  jwsitive  quadratic  surd,  or 
any  binomial  quadratic  surd  in  ivhicli  one  term  is  rational  may 
he  expressed  as  a  recurring  continued  fraction. 

EXAMPLES 

1.    Reduce  V5  to  a  continued  fraction. 

The  greatest  integer  in  V5  is  2. 

Hence  we  write  V5  =  2  +  ( V5  —  2)  =  2  + 


V5  +  2 
Again,  the  greatest  integer  in  Vo  +  2  is  4. 

Hence       V5  +  2  =  4  +  ( V5  -  2)  =  4  +  —J — , 

V5  4-2 

where  the  fraction  to  be  further  converted  is  the  same  as 
before. 

Hence  V5  =  2-hi:      i      ...=(2,4). 

4  +  4  + 

As  a  verification  of  the  results  we  employ  the  methods 

°''"'-  ^  o_      1      _  1 

4  + (2:- 2)      x^-2 

Hence  x^=  5,  and  x=±  V5» 

and  X  =  Vo  is  a  root  of  the  equation. 

2.    Reduce  VT  to  a  continued  fraction. 


VT  =  2  +  (  VT  -  2)  =  2  +  —S —  =  2  + 


1 


V7  +  2  V7  +  2 

3 


3  3(V7  +  1)  V7  +  1 


192  COLLEGE   ALGEBRA 

V7  +  1      ,    ,   V7-1      .    ,  6  ,1 

— - —  =  lH —  =  lH .  =  iH ^ 

2  2  ^2(V7-1)  V7  +  1 

3 
V7  +  l^-|+V7-2^-^  ,  3  ^1,        1 

3  3  3(V7  +  2)  V7  +  2 

V7  +  2  =  44-(V7-2). 
Therefore  VT  =  (2,  1,  1,  1,  4). 

Let  the  student  verify  the  result. 

3.    Reduce  V47  to  a  continued  fraction. 
V47  =  6  +  (V47-6)=6  + 


V47  +  6 


11 


V47  +  6      .      V47-5      ,  1 


11  11  V47  +  5 

2 

V47  +  5      .      V47-5      .  1 

11 

V47  +  5_^  ,  V47-6_^  ,         1 


11  11  V47  +  6 

V47  +  6=12  +  (V47-6). 

Hence  V47  =  (6,  i,  5,  1,  12). 

It  will  be  observed  that  the  last  quotient  before  the  period 
begins  to  recur  is  twice  the  initial  quotient,  and  that  within 
the  period  there  is  a  reverse  recurrence  of  the  quotients.  A 
general  proof  of  these  properties  could  be  given.  They  are 
of  value  in  checking  the  accuracy  of  the  computation. 


CONTINUED   FRACTIONS  193 

Convert   the    following   quadratic    surds    into    continued 
fractions,  and  verify  the  results : 

4.  Vn.  6.    V21.  8.    VT9. 

5.  Vl7.  7.    Vl8.  9.    2-fV3. 

10.  Show  that  2  -  V8  =  (0,  3,  1,  2). 

Convert  the  following  surds  into  continued  fractions,  and 
verify  the  results : 

11.  5-V2.  13.    7-V7.  15.    ^^11^. 


5 

14.    2+V3  7_V6 

lb. 


12.    3-V5.  ^«.    

^  3 

243.    Forming  the  differences 

^  — ^= ,  etc., 


(236) 


and  denoting  by  m  that  value  for  which  this  has  been  ob- 
served to  hold,  we  have 

Pm        Pm-l^j-'^T  _ 
9.m         9m -I  9.m-\2m 

•g^^  Pm_  _  Pm-1  __  Pm9m-l  ~  Vm-l^m  ^ 

9.m         9.m-\  %n-\9.m 

Therefore  the  numerators  of  the  second  members  must  be 
equal,  or  _  —(_\yn 

Thpn    Pm-^\  Pm ^m+-[2^m  "^  Pm-\  Pjn  Pm-\  ^w        Pm9.m-\ 

9.m+l         %n         ^m+l9m  "r  9m-l  Qm  ^m^m+l 

=  — ^ ^.         (237) 

9m9m+-i 


194  COLLEGE   ALGEBRA 

Therefore  P'n+i9m- Pmq>n+i  ^  (-1^^^ ^ 

Hence  the  law  holds  for  a  value  m  +  1.  Since  it  holds  for 
m  =  3,  it  holds  for  w  =  4,  and  so  on  for  all  positive  integral 
values.  Hence  the  law  Pm9.m-\— Pm-i%n=  (~1)'"  is  true  in 
general. 

244.  Every  convergent  —  is  in  its  lowest  terms,  for 

and  therefore  p,^^  and  q,^^  have  no  common  ^factor  but  unity. 

245.  Conver gents  are  alternately  greater  and  less  than  the  con- 
tinued fraction,  and  each  one  is  nearer  to  it  than  the  preceding. 
The  first  convergent  a^  is  too  small,  the  second  convergent 

a^-\ —  is  too  large   because  a^  is   too  small,  the  third  con- 
H  \  1 

vergfent  <^o  H t  is  too  small  because  ao  is  too  small,  or  — 

^  'i'  '  \  ^  a^ 

^2  +  - 

is  too  large,  and  —  is  therefore  too  small,  and  so  on. 

«2  +  — 


P 

Denoting  —  by  x, 


H 


X 


Pm  ^  -gpm+i  -f-  P,n        Pm  ^         -^(  -  ^Y^^ 
<lm         Kqm^^-[-q„,  q„,         qm(^q7n+l  +  qmY 


where  K=  a.,,^^-^  + 


1  1  1 


X  — 


^m+2        ^'^in+S    '  '    ^n 

Pm^-\  Pm^m+l        Pm+l9m  \        -'■/ 


qm+1     qm+ii^qm+1+qm)    qm+iiKqm+i  +  qm) 


CONTINUED   FRACTIONS  195 

These  relations  show  that  two  consecutive  convergents  lie  on 
opposite  sides  of  the  value  of  the  continued  fraction,  and 
since  K is  not  less  than  1,  and  q,^^^  =  a,^^^q„,  +  q,n-i>qm.  it  is 
clear  that  the  second  difference  is  numerically  less  than  the 

first,  and  hence  ^^  is  nearer  to  x  than  —  • 

9'm+l  qm 

The  error  in  taking  —  for  x  is 


^<^-l)"'"'    ...    (-1) 


WJ  +  1 


or 


qm(J^qm+i-\-qny      ^,  (^     _,q,n 

q»i[qm+i  -r  -^ 


Since  —  >  — — 


qm    qmqm+i 

11                                  11 
we  have  --  > >  e^>  - — > 


qn?     qviqm+1      ""    qmOim+i  +  qm}    ^q,    ^ 


[m+l 


where  e,^  =  — y denotes  the  numerical  value  of  the 

error.     — -  and are  looser  limits^  and and 

qm         ^  qm+i  qmq>7i+i 

are  closer  limits  of  error. 


qm{qm+i  +  qm) 

EXAMPLES 

1.    Find  the  closer  limits  of  error  in  taking  its  5th  conver- 
gent for  ||l 

The  5th  convergent  is  |1,  qQ=  244. 

Hence  the  error  lies  between  — —  and 


41  x244  41(244+41)' 


or  between  and 


10004  11685 

2.   Find  the  closer  limits  of  error  in  taking  the  5th  conver- 
gent for  f  Jf f . 


CHAPTER  XIV 

INTEGRAL  SOLUTIONS  OF  INDETERMINATE  EQUATIONS  OF 

THE  FIRST  DEGREE 

246.  Since  by  243  of  the  previous  chapter,  Pn<ln-i—  Pn-\% 
=  (—!)%  it  is  easily  seen  that  solutions  of  the  equation 
ax-\-  by  =  \  can  be  found  when  a  and  h  are  prime  to  each  other 
if  we  put  h  =Pn->  ^  =  5'n  ii^  ^^e  preceding  relation  and  choose  for 
X  and  y  the  numerical  values  of  j9„_j  and  qj^_^  with  such  signs 
as  shall  satisfy  the  equation  ax-\-hy—  1.  jt?„_j  and  qn_^  are 
the  terms  of  the  next  to  the  last  convergent  in  the  process  of 

converting  -  into  a  continued  fraction.     If  x^  and  y-^  denote 
a 

the  solutions  so  obtained^  the  general  solution^  that  is,  formulas 
which  contain  all  integral  solutions,  are  evidently 

x  =  x^-\-  rb,         yz=z  y^  —  ra, 

where  r  is  any  positive  or  negative  integer  or  zero.     For,  we 

have  7  -,         1  7        -I 

ax-^  +  oy-^  =  1,  and.  ax  -\-  by  =  1, 

and  therefore  a(x-^  —  x^-\-  b(y^  —  ^)  =  0, 


or 


iCj  —  a;       —  b 


y\-y      ^ 

whence  x^  —  x=  —  rb,  and  y^  —  y  =  ra, 

or  x  =  x■^^  +  rb,  and  y  —  y^  —  ra. 

100 


INTEGRAL   SOLUTIONS  197 

Example.    Solve  the  equation  llx  +11 1/  =  1. 

17  12 

We  find  —  =  (1,  1,1?  5),  and  the  convergents  are  -,  -^ 
3    ^  11  11' 

2'  11' 

Therefore  if  we  choose  x  =  —  S,  and  ^  =  2,  the  equation  is 
satisfied.  Then  the  general  solution  is  x  =  —S  +  llr  and 
^  =  2-llr. 

247.  Theorem.  If  x^  and  y^  constitute  a  particular  solution 
of  the  equation  ax-{-bi/=  1,  theii  cx^  and  cy-^  constitute  a  par- 
ticular solution  of  the  equation  ax+hy  —  c,  and  the  general 
solution  of  this  equatio7i  is 

X  =  cx^  +  ^^  ciyid  y  =  cy-^  —  ar. 

If  a  and  h  have  the  common  factor  t?,  that  is,  if  a=  da\  and  h 

=  dh\  then  a'x  -\-b'y  =  -  and  therefore  no  integral  solutions  of 

d 

this  equation  are  possible,  since  the  right-hand  side  is  a 
fraction. 

EXAMPLES 

1.  Find  the  general  solution  in  integers  of  13a;  —  19^  =  1. 

19  1   3    19 

We  have  —  =  (1,  2,  6),  with  convergents  j'  9'  tTj'    Hence 

particular  values  of  x  and  y  are  3  and  2,  and  the  general  solu- 
a;  =  3  +  19  r  and  y  =  2  +  13  r. 

2.  Find  the  integral  values  of  x  and  y  which  satisfy  the 
equation  bx-]-  S  y  =  37. 

Q  1    '^    3    8 

-=   (1,  1,  1,  2)  ;  the  convergents  are  -,  ^,  -,  -, 
5  1    1    J    o 

therefore  for  the  equation  5  a;  -f  8  ^  =  1,  a:^  =  —  3,  y^  =  2. 

cx^= —111,  cy-^=14:. 


198  COLLEGE   ALGEBRA 

Hence  the  general  solntion  is 

a;=-lll4-8r=l-112  +  8r=l-  8(14  -  r), 
^  =  74  -  5  r  =  4  -1-  70  -  5  r  =  4  +  5  (14  -  r), 

or,  denoting  14  —  r  by  s,     x=l  —  Ss^ 

«/  =  4  +  5  s, 

where  s  is  zero,  or  any  positive  or  negative  integer.  We  note 
that  the  equation  has  but  one  solution  in  positive  integers, 
namely,  for  s  =  0.  It  may  also  be  observed  that  if  we  can 
see  by  inspection  any  special  solution  of  our  equation,  we  can 
at  once  write  down  the  general  solution.  Thus  in  the  pre- 
ceding example,  if  we  had  seen  that  1  and  4  were  values  of 
X  and  ^  satisfying  the  equation,  then  the  general  solution 
could  have  been  written  down  at  once. 

248.  EXAMPLES 

Find  the  general  integral  solutions  of  the  following  equa- 
tions : 

1.  53  a: -19^  =  1.  5.  7  a; +  9?/ =  4. 

2.  23  a; +5?/  =  1.  6.  5  a; -11?/ =  27. 

3.  37a:-29y  =  19.  7.  14a;-33?/  =  49. 

4.  11 2; +  41  ^  =  17.  8.  3  a: -25?/ =61. 

9.  Two  bells  begin  to  ring  together.  One  rings  eleven 
times  in  five  minutes,  and  the  other  thirteen  times  in  seven 
minutes.  What  strokes  most  nearly  coincide  in  the  first 
fifteen  minutes? 

Note.  This  problem  is  one  illustrating  the  use  of  convergents  and 
not  indeterminate  equations. 


CHAPTER   XV 
SUMMATION  OF  SERIES 

249.  We  have  already  seen  how  to  find  the  sum  of  certain 
types  of  series,  namely,  arithmetical  and  geometrical  pro- 
gression and  some  simple  integral  series  such  as  those  of 
232,  233. 

250.  Let  us  start  with  the  series  whose  nth  term  is 

UJ^=  (^a  -\-  nh^ [a  +  (?^  +  l)i]  •  •  •  [a  -f  (^i  +  m  —  1)^] , 

where  it  is  to  be  observed  that  the  m  factors  of  any  term 
are  in  arithmetical  progression,  and  where  the  correspond- 
ing factors  of  any  m  consecutive  terms  of  the  series  are  con- 
secutive terms  of  the  same  arithmetical  progression.  The 
sum  of  n  terms  of  this  series  may  be  found  as  follows : 


'n' 


Let  v„  =  [a  +  (?i  +  m)J]w, 

then  v^_j  =  [a  +  (w  +  yn  —  l)J]?^„_-j  =  \_a-\-  (n—  l)5]w„. 

Hence  v^^  —  i;„_j  =  (m  4-  l)5w„, 

and  changing  n  into  ?i  —  1,  n  —  2,  etc., 


^2  —  ^1=  0^^+  1)^"2' 

^1  ~  ^'o  —  0^^  +  lyhu^. 
199 


200  COLLEGE  ALGEBRA 

Adding   the  corresponding  members  of   these  equations, 
we    ave   ^^  _  ^^ ^  ^^^  ^i^^(^u^-}- u^  +  u^-h  "•  +  w„) 

n 

where  2  Uy  =  u-^^ -\- u^  +  u^ -\-  •  •  •  +  w„, 

which  is  the    sum  of   the  expressions  of   the    form  Uj.  as  r 
assumes  all  positive  integral  values  from  1  to  7i,  inclusive. 


Therefore  ^u  =    ^»  ~  '^o 

1    ''      (m-\- 1)6 


EXAMPLES 

1.  Find  the  sum  of  n  terms  of  the  series 

1.2  +  2. 3  +  3. 4+---. 
Here  the  nth  term  u^  =  n(^7i  +  1). 

Hence    zUj.  =  — — — — = -n{n  +  l)(n-{-2). 

1  3  3 

2.  Find  the  sum  of  n  terms  of  the  series 

3.5  +  5.7  +  7.9  +  .... 
Suggestion  : 
w„  =  (l  +  2n)[l+2(n+l)]  =  (2  n  +  1)(2  n  +  3). 
y„  =  (1  +  2  n)[l  +  2(n  +  1)]  [1  +  2(n  +  2)]  =  (2n  +  l)(2n  +  3)(2n  +  5). 

251.    Let  us  now  consider  the  case  when  the  nth.  term  is 
the  reciprocal  of  that  considered  in  250,  viz., 

1 

"      (a  +  n^>)[a+  {n  +  l)b]  .-.[«+  (71  +  771  -  1)5] 

Let  v^^  =  (a  +  7ih')7i„^ 

then  v.n_^  =  [a  +  (?i  —  l)5]i/,,^_i  =  [«  +  (n  +  m  —  l)5]t^„. 

Hence  v,^  —  i'„_j  =  —  (m  —  T)hu,^. 


SUMMATION   OF   SERIES 


201 


Changing  n  into  n—1,  7i  —  2,  etc., 


v^  —  Vq=  —  (m  —  l^hu^. 


Adding  the  corresponding  members  of  the  several  equations, 


or 


1  (772-1)6 


Ur 


Note.  It  is  to  be  observed  that  this  formuLa  for  the  sum  of  n  terms  is 
the  same  as  that  of  the  preceding  section  when  7n  is  changed  into  —m,  but 
it  is  not  to  be  supposed  that  one  series  can  be  obtained  from  the  other  by 
changing  m  into  —  m.     In  fact  in  each  series  7n  must  be  a  positive  integer. 

252.  EXAMPLES 

1.  Given  u„  =  n(7i  -\-  l)(w  +  2).     Find  the  sum  of  n  terms. 

2.  Given  u,^  =  n(ii  +  2)(?i  +  4).     Find  the  sum  of  n  terms. 
Suggestion  :      u^  =  ?<[(«  +  1)  +  1]  \_{n  +  2)  +  2] 

=  ;;[(n +  !)(«  + 2)  +  2(n  +  1)  +  (n  +  2)  +  2] 
=  n(n  +  1)  (n  +  2)  +  3  7i(n  +  1)  +  3  n. 

Thus  iij^  is  resolved  into  three  parts,  each  of  which  can  be 
treated  by  250. 

Find  the  sum  of  n  terms  of  each  of  the  following  series : 

3.  1.2.4  +  2.3-5  +  8.4.6+-.-.     . 

4.  1.4. 7  +  4. 7-10  +  7. 10.13H-.... 

5.  1.7  +  2.8+3.9+.... 

6.  1.7  +  3.9  +  5.11+....  8.     I2  +  22+.32  +  .... 

7.  1.22+3.42+5.02+....  9.     124-32  +  52+..., 


202  COLLEGE   ALGEBRA 

10.  Show  that  12  -  22  +  32  -  42  -f  ...  +  (2  w  +  1)2 

=  (7i  +  l)(2n  +  l). 

11.  Show  that  12  _  22  +  32  -  42  + (2  n)'^ 

=  —  w(2  n  +  1). 

Find  the  sum  of  n  terms  of  each  of  the  following  series : 

12.  13  +  23  +  33+.... 

Suggestion:    n^=n(n  +  l)(n  +  2)  -  3  w(n  +  1)  +  n. 

13.  14  +  24  +  34+  .... 

14.  _l__+_J_+^+.... 

1.2-3      2.3.4      3-4-5 

15.  — = 1 = 1 \ . 

1.3-5      3.5.7      5.7-9 

16. \ \ = \ . 

1.2.42.3.53.4-7 

Suggestion  :   w„  —  —  ^ 


n  {n  +  1)  (n  +  3)       n  (n  +  1)  {n  +  2)  (n  +  3) 


(n  +  l)(/i  +  2)(n  +  3)      n(n  +  l)(n  +  2)(n  +  3)  * 

17. +— — ^ + +  .... 

1-3. 5. 73. 5. 7-95-7-9- 11 

Suggestion:     n^  =  i(2  w  -  l)(2n  +  1)  +  ^. 

18.     = 1 = 1 1 . 

1-2-5      2-3-0      3-4-7 

19.  Find  the  number  of  shot  in  a  pyramid  with  a  trian- 
gular base  with  40  shot  on  a  side. 

20.  Find  the  number  of  shot  in  a  pyramid  having  a  square 
base  with  50  on  a  side. 


SUMMATION  OF  SERIES  203 

21.  Find  the  number  of  shot  in  a  wedge-shaped  pile  with 
a  rectangular  base,  the  lower  layer  containing  21  on  one  side 
and  14  on  the  other,  the  next  layer  with  20  on  one  side  and 
13  on  the  other,  the  top  layer  being  a  single  row. 

22.  Find  the  number  of  shot  in  a  wedge-shaped  pile  of  the 
same  sort,  of  which  the  lower  layer  contains  m  shot  on  one 
side  and  n  on  the  other,  where  m  is  greater  than  n. 

RECURRING  SERIES 

253.  Definitions.  A  recurring  series  is  a  series  in  which,  be- 
ginning ivith  and  after  a  certain  term,  each  term  is  equal  to  the 
sum  of  a  fixed  number  of  the  2^receding  terms  multiplied  re- 
spectively by  certain  constants.  A  recurring  series  is  of  the 
first,  second,  third,  etc.,  order  according  as  the  fixed  number 
of  terms  is  one,  two,  three,  etc.     Thus  in  the  series 

l^^x^^x^  +  4:X^-\ hW2;"-iH , 

naf-i  =  2  x(^n  -  l).T"-2  -  x\n  -  2)a:"-3, 

or  nx""-^  —  2  x(n  -  l^x"'^  +  a^(n  —  2)x''-^  =  0,  if  n  >  2, 

and  the  series  is  of  the  second  order.  In  this  relation  the 
multipliers  which  are  constant  with  respect  to  n  are  1,  —  2x, 
and  x^.  Their  sum,  that  is,  the  expression  1  —  2x-{-x^,  is 
called  the  scale  of  relation  for  the  series  and  in  general  if  the 
Uq-{-  u^x  -j-  ti^x"^  -{-•■•+  iiyiX^  +  •••  is  a  recurring  series  of  order 
r,  and  if 

UnX"^  ■{-  P'^XUn-iX^~'^  +  j02^W'«-2^"~^  + h i?r^'''?^n-r^" ~ '"  =  ^» 

the  expression  1  -^  p^x-\-p^x^-\-  •••  +  p^x^  i^  the  scale  of  relation 
for  the  series.  When  the  constants  p^,  p^,---,p,.,  are  known, 
that  is,  when  the  scale  of  relation  is  known,  the  series  is 
determined. 


204 


COLLEGE   ALGEBRA 


254.  If  we  are  given  any  2  r  consecutive  terms  of  a  series 
of  order  r,  we  can  find  the  scale  of  relation,  because  we  are 
able  to  form  r  independent  equations  between  the  r  constants 
of  the  scale.  Thus  to  find  the  scale  of  relation  of  the  series 
of  the  second  order, 

we  have  11  x^  -{-  'px  %x-\-  qjr  =  0, 

and  43  x^  -\-  px  11  x^  +  qx^  3  =  0, 

or  11-{_3^_I_,^^0,                          (1) 

and  43  +  lljt?  +  3^=:0.                         (2) 

Subtracting  3  times  (1)  from  (2),    10  +  2  |?  =  0, 
or  p  =  —  5^ 

and  therefore,  ^  =  4. 

Hence  the  scale  of  relation  is  1  —  5  x  +  4:  x^.  By  using  this 
we  can  determine  as  many  more  of  the  terms  of  the  series 
as  we  please.     Thus 

UnX^  —  5  xUn^-^x'^~'^  +  4  x^u^i_2X^'~'^  =  0, 

or  w„  =  5  Un_-^  —  4  ^t,^_2,  Avhen  71  >1. 

Hence,  u^  =  215  —  44  =  171, 

^,^  =  855-172  =  683, 

u^  =  3415  _  684  =  2731, 

and  the  next  three  terms  are  171  x^^  683  x^^  2731  x^. 

255.    To  find  the  sum  of  n  terms  of  a  recurring  series  we 
proceed  as  follows :   Taking  the  series  of  order  two, 


6^  —  Un  ~p  U-l  X  ~\~   ttnX     ~\~ 


~r  U^i_-^x      , 


(l; 


pX8,,  =  PUqX  +  pU^x"^  +   •  •  •    +  pUn-2^''~^  +  pUn-i^'^^ 


qx^s^  ■—  qu^x^  -f- 


SUMMATION   OF  SERIES 


205 


Adding,  we  liave 
(1  -\-  2)x  -{-  qx^)s,^ 

since  u,.-\- 2^^,-1  +  ?^;-2  —  ^»  when  ?- >  1. 

Hence, 

_  '?<Q  +  (u^  +  /;^o)a;  +  (pun~i  +  qif'n-2)^'"  +  g^n-r^""^^ 


(2) 


256.  For  values  of  x  which  make  the  series  convergent,  it 
is  necessary  that  L  u^x^  =^^  by  196,  Cor.    1.     Taking  the 

72=00 

limits  as  n  becomes  indefinitely  great  of  both  members  of 

Equation   (z),  we   have   s  = -^ — ^^^^ — ^ 1^,   since    M„_ja;\ 

Un-2^^\  Un-yc'^'^^^  may  be  written  xUn-iX^~^,  x^Un_2X^~^^  x^Un_^x^~'^^ 
respectively. 

257.  Ihe   expression   -^^-:^ — - — ^ ^   is   the   generating 

function  of  the  series.     This  is  obvious  because  it  is  tlie  sum 
of  the  series,  and  the  series  may  be  reproduced  as  follows  : 


2:"  +  ••-. 


1  -\-px-\-  qx^           "^ 

-  u-iUy  -\-  v.- 

yt        T       •■ 

•  T-  ^)i^   ^ 

Then 

u^  +  (pi(o  +  yi)x  =  v^  +    ^1 

x-h    V^ 

a;2+  . 

••    +      Vn 

+  7^i'o 

+  pv^ 

+  PVn-l 

+  ro 

+  qVn-2 

Equating  coefifiicients,  Vq  =  Uq,  pv^  -\- 1\  =  jj»?^q  +  i^j,  and  z'j  =  Wj, 
V,. +pv^_j  +  (yi'^_2  =  0,  r>l,  and  this  is  the  relation  which 
gave  the  successive  terms  beyond  u^  in  the  original  series. 


206  COLLEGE   ALGEBRA 

258.  EXAMPLES 

1.    Find  the  scale  of  relation  and  the  generating  function 
of  the  series  1  -\-  2  x  -{-  5  x^  -j-  SI  x^  -\-  •••. 

Find  the  generating  function  of  each  of  the  following 

series : 

2.    2-{-9x-\-25x'^  +  66a^-\-  --'. 

„     5  ,  3a;      21  x'^  ,  45  x^  . 

4.  -S-^-Sx-lSx^-i-lSx^-  '--. 

5.  2-9x-{-SSx^-lllx^+  -". 

6.      S  —  X  —  4:X^-\-Sx'^  —  X^—    •■■. 

7.  1- Sx-{- 6x^-10:}^ +  15  x^- 21  x^+  '". 

8.  1  +  22  2:  +  32  a;2  +  42  x^  +  52  x^  +  (j^  x^  +  .... 

9.  l-Sx-]-5x^-7x^+  .... 

10.  Find  the  nth  term  of  the  series 

_3_11U_17£^_39^_ 

2        4  8  16 

11.  Find  the  71th  term  of  the  series 

3  _  19^      113^  _  679^ 

2       12  72  432         "'* 

12.  Find  the  sum  of  n  terms  and  the  generating  function 
of  the  series  Uq  +  u-^x  +  u^x'^  +  •••  +  Un_-^x^~'^  +  •••?  whose  scale 
of  relation  is  1  -{-  p^x  -{-  p<^x^  -{-  p^oi^. 

FINITE   DIFFERENCES 

259.    If  the  terms  of  a  series  are  iIq,  u-^^  u^,  7/3,  .••,  u„^  ..., 

then  let  A?/q,  A?/j,  H^u^,  •••,  Ai6„_j,  •••,  denote  the  differences 


SUMMATION  OF   SERIES  207 

A^Wq,  A^z/j,  ^^u^,  '••,  denote  the  differences 
Awj  —  Awq,    Awg  —  Amj,    Ai^3  —  Aw2, 
and  in  general  let 

A^?fo=A^-i?^i-A^-iMo,  A^i^i  =  A''-iw2  -  ^"""^^^  •••• 

The  differences  A?*^,  A^?/;^,  A'^%.,  •••,  A^%.,  are  called  the  differ- 
ences of  the  firsts  second^  third,  •••,  rth  order  respectively. 
A  may  be  considered  as  an  operator,  and  that  it  is  distribu- 
tive is  seen  from  the  following: 

=  U^+^  —  lij.  +  llk+i  —  Uk  =  ^Ur  +  A%. 

Also  A(Aw,)  =  A^w,. 

260.    Having  given  a  number  of  terms,  we  may,  by  find- 
ing the  differences,  find  the  general  term. 

By  259  we  have       Uj,+i  =  Ur  +  Aw^. 
Thus,  Uj^  =  Wq  +  Auq, 

u^  =  Ui  H-  Au^  =  i<Q  +  Ai^Q  +  ^('^0  +  ^'^o) 
=  Hq  +  Auq  +  Ai<j)  +  A\, 
or  ^2  =  ?^Q  +  2  A?<^^  +  A^z^Q. 

Let  m  be  a  value  for  which  it  has  been  observed  that  the 
following  law  is  true  : 


0    '    \   -1    j'-"0 


0 

_L  /  "Maw-,/  . 


m 


+  (  "'  W% 


208  COLLEGE   ALGEBRA 

Since  u^+i  =  w^^  +  ^^m^  we  have 

»«.:  =  (;)"o  +(7)  A»„  +g)A^«„  +  ...  +  (-)a-.„  +  ... 

\m  —  1/  V^^i/ 

or,  remembering  that  f      )  +  (         -,  )  — (  )  ^^^^^  ^^^^^  (      j 


0     /         \7nJ     W  +  iy 


'm  +  1 


and  hence,  by  the  principle  of  mathematical  induction,  the 
law  is  true  in  general. 

We  may  write  the  above  symbolically  in  the  form 

u,,  =  (1  +  AT'Uo, 
which  gives  the  (w-f-l)th  term. 

261.    The  sum  of  m  terms  may  be  found  as  follows  : 

We  have  v^^  =  ?/q, 

u^  =  u^  +  Auq, 

u^  =  Mq  +  2  A?/q  +  A^Wq. 

Hence,  Sg  =  k.^  +  ?/i  +  ti2  =  3  iCq  -\-  3  A?^^  +  A^Uq. 


SUMMATION    OF   SERIES  209 

Let  m  be  any  value  for  which  it  has  been  verified  that 

«».=(7)«o+©^»o+-+(:)a-x+-+(:)a»-%„. 

Then  ,    .  .    .  , 


+ 


Therefore  the  law  holds  for  a  value  of  m  one  greater.  Since 
it  is  true  when  m  =  3,  it  is  true  for  m  =  4,  5,  •••,  that  is,  for 
all  values  of  m.     We  may  write  symbolically, 


8  m    


A 


%• 


262.  EXAMPLES 

1.    Find  the  wth  term  and  the   sum   of  m  terms  of  the 
^^^^^^  2  +  2+8  +  20  +  38  +  .... 

Finding  the  differences  of  the  various  orders,  we  have 

2         2  8  20         38 

0  6  12  18 

Q  Q  Q 

0  0 

That  is,  Wq  =  2,  A?(,3  =  0,  Ahi^  =  6,  A%  =  0  when  r  >  2. 

Note.     That  A'«o  =  0  for  all  values  of  r  greater  than  2  follows  from  the 
assumption  that  a  sufficient  number  of  terms  have  been  given  to  determine 
a  law  which  the  given  series  will  obey. 
p 


210  COLLEGE   ALGEBRA 

The  (m  +  l)th  term  is 

%n  =  -  H Hs 6  =  2  +  3  m  (m  —  1). 

Hence  u„^_^  =  3  (w  —  l)(w  —  2)  +  2. 

Alsos„,  =  m2H ^ 4^^ ^b=  2m  +  m(m  — l)(m  — 2). 

2.    Find  the   mth  term    and  the  sum  of   m  terms  of  the 
series 

1  +  4  +  11  4-  26  +  57  +  120  +  .... 

We  have  as  before  the  series  of  differences 

1       4       11         26         57         120 
3       7         15         31         63 
4         8         16         32 
4  8         16 

4         8 
4 

Here  u^  —  l^  ^u^  =  3,  A^'w^^  =  4  for  all  values  of  r  greater 
than  1.      (See  note  on  previous  problem.) 

Hence 


l^m    ■        _ 


(:>H-(T)^+(:>-(;>+- 


+  (  ^  U  _  3f  ^M  _  n'^  ]  =  4  .  2'«  -  3  -  m  =  2'»+2  _  3  _  ^^^ 


SUMMATION   OF   SERIES  211 

Also  u^^;^  =  2"^+!  -  2  -  m. 

+(s>+-+(:>-<:)-<T)-(2)=^"^^-^ 

263.  If  we  are  given  but  a  few  terms  of  a  series  without 
being  given  the  law  of  the  series,  we  may  determine  a  law 
which  the  given  terms  will  obey,  but  it  is  to  be  understood 
that  this  may  not  be  the  only  law  which  these  terms  obey, 
for  it  may  be  possible  to  have  the  given  terms  obey  different 
laws.     For  example,  the  series 

treated  by  the  method  of  differences  has  for  its  nth  term 
(Sn^—9  7i-\-  8)a;"~i  as  is  seen  from  Example  1  of  262.  The 
same  series  considered  as  a  recurring  series  of  order  two  has 

2—  2ic 
a  generating  function — ^  which  yields  as  a  gen- 

T      .  JL  —  A  X  —  jJ  X 

eral  term 

a  —  p 

where  a  and  fi  are  the  reciprocals  of  the  roots  of 

l-2a:-2a.^=0. 

It  is  seen  that  the  coefficient  of  x^  is  a  function  of  sym- 
metric functions  of  a  and  ^8,  and  it  can  be  proved  that  such  a 
function  is   rational,  but  it  is   more   complicated   than   the 


212  COLLEGE   ALGEBRA 

coefficient  of  the  general  term  obtained  by  the  method  of 
differences.     It  satisfies  the  recurrence  formula 

Hence  the  coefficient  of  x'^  by  it  would  be  3^^  while  from  the 
former  standpoint  it  would  be  3  •  5^  —  9  •  5  +  8  =  38. 

Thus  the  two  series  agree  to  the  fourth  term  but  diverge 
after  this. 

264.  EXAMPLES 

Find  the  mth  term  and  the  sum  of  m  terms  in  the  follow- 
ing series: 

1.  2-f-3  +  7  +  16  +  324----.         4.   3  +  8 +  10  +  10  +  9  +  •••. 

2.  Y  +  8H-11  +  17  +  31  +  62+....     5.   2  +  5  +  10  +  9  +  6 +  •.-. 

3.  _2  +  0  +  l  +  4  +  12+....        6.   3  +  2-1  +  2  +  3-22  +  .... 

7.  _4  +  o  +  18  +  5G  +  120+--.. 

8.  2  +  5+9  +  15  +  25  +  43+--.. 

9.  4  +  7  +  12  +  21  +  38  +  71  +  .... 
10.  1  +  3  +  8  +  19  +  42  +  89  +  .... 

11.  Find  by  the  method  of  finite  differences  the  (m  +  l)th 
term  of  the  series  2  +  5a;  +  9:c2  +  15a;^  +  25a;*  +  43a:;^+.... 

INTERPOLATION 

265.  If  we  have  a  series  of  terms,  f(x),  f(x  +  1),  /(a;  +  2), 
•  .  •,/(:r  +  n),  we  may  put 

f(x  +  1)  =  Wj, 
f(ix  +  2)  =  u^, 

fQx  +  7l)  =  Wn- 


SUMMATION    OF   SERIES  213 

Whence,  by  260, 

f(x  +  n)  =  (1  +  i^rfix)  =/(a.)  +  Qa/(x) 

This  formula  has  been  proven  for  the  case  where  7^  is  a  posi- 
tive integer.  The  development  given  iov  fQx  +  n)  can,  how- 
ever, be  proven  to  hold  for  all  real  values  of  n  which  make 
the  development  convergent.  It  is  often  used  as  a  formula 
for  interpolating  values  of  a  function  between  values  corre- 
sponding to  integral  values  of  x.     Thus,  given 

/(I)  =  2, 
/(2)  =  5, 
/(3)  =  10, 

/(^)  =  17, 

to  find  /(l^).     In  this  case  x=l^  and  we  have 

A/(l)  =  3, 
A/(2)  =  5, 
A/(3)  =  7, 
Ay(l)  =  2, 
A2/(2)  =  2, 
A3/(l)  =  0, 

A:/(1)  =  0,  r>2. 


214  COLLEGE   ALGEBRA 


Therefore 


/(1|)  =/(l)  +  I  A/(l)  +  J(f      ^)a2/(1) 

_  9  _l_  9 3_  _  6  5  _  4JL 

"""^4         16~~16~^16* 

As  a  verification  we  have  the  {n-\-  l)th  term, 
f(n  -H  1)=/(1)4-  Qa/(1)  +  QaVCI) 

when  ?^  is  a  positive  integer. 

Put  w  +  1  =  a:,  or  n  =  x  —  \^ 

then       /(a:)  =  2  +  3  (a:  -  1)  +  (a^  -  1)  (a:  -  2)  =  2;2  +  1, 

which  is  true  for  more  than  two  positive  integral  values  of  x 
and  therefore  for  all  values  of  x.     Whence 

f(\^\  =  49    I    1  _  65  —   IJL 
J  \^i^)  —  16  ^  ^  —  16         ^16* 

As  another  example,  given 

log  50  =  1.698970, 

log  51  =1.707570, 

log  52  =  1.716003, 

log  53  =  1.724276, 
to  find  log  50.13. 

Here  x  —  50,  and  we  have, 

log  50.13  =  log  50  +  (.13)  A  log  50  +  (-^3)0^^-^)  a2  log  50 

(.13)  (.13-1)  (.13 -2)  ^3  log  50. 
3! 


SUMMATION   OF   SERIES  215 

This  is  as  far  as  our  data  extend,  since 
A  log  50  =.008600, 

A  log  51  =  .008433, 

A  log  52  =  .008273, 

Anog  50  =  -.000167, 

A2  log  51  =  -.000160, 

A3  log  50  =  .000007. 
Hence 
log  50.13  =  1.698970  +  (.13)  (.008600)  +  C-^^X-^^-l) 

(-.000167)  +  (■1S)C1'^-^)C1^--)  (.000007)  =  1.700098.* 

3 ! 

It  may  be  noticed  that  the  second  term,  (.13)Alog50,  is 
the  amount  added  to  log  50  by  the  usual  interpolation  to  find 
log  50.13.  The  succeeding  terms  furnish  additional  correc- 
tions. 

266.  EXAMPLES 

1.  The  cube  roots  of  60,  61,  62  are  respectively  3.01587, 
3.93650,  3.95789;  find  the  cube  root  of  60.25. 

The  positions  of  a  comet  at  Greenwich  mean  midnight  are 
as  follows  : 

Eight  Ascension  Declination 

h       m         s 

March  15,  1907  6  48  27  - 12°  46' 

March  19,  1907  6  40  10  -  9      28 

March  23,  1907  6  33  18  -  6      26 

March  27,  1907  6  27  44  -  3      42 

*  What  function  of  a  number  a  logarithm  (log)  is  will  be  explained  in  the 
next  chapter. 


216 


COLLEGE   ALGEBRA 


What  is 

2.  Its  right  ascension  at  Greenwich  mean  midnight  on  the 
16th  of  March,  1907  ? 

3.  Its  declination  at  Greenwich  mean  midnight  on  the  16th 
of  March,  1907? 

A  rifle  was  shot  at  different  ranges  and  the  following  table 
of  elevations  e  for  the  vernier  peep  sight,  for  the  given  dis- 
tances d  was  obtained. 


d 

e 

0 

21.0 

100 

24.5 

200 

28.5 

300 

33.5 

400 

40.0 

500 

48.5 

4.  What  is  the  elevation  for  425  yards  ? 

5.  Solve  generally  and  find  what  function  of  the  distance 
the  elevation  is. 


CHAPTER   XVI 
LOGARITHMS 

267.  Definitions.  If  we  consider  the  equation  a^  =  ?/,  the 
problem,  given  two  of  the  three  numbers,  a,  x^  ?/,  to  find  the 
tliird,  leads  to  the  consideration  of  the  following  types: 

1.  Given  x  and  y^  to  find  a. 

2.  Given  a  and  a;,  to  find  y. 

3.  Given  a  and  y^  to  find  x. 

The  solution  of  the  first  presents  itself  in  the  form  a  =  v^, 
by  taking  the  2:th  root  of  both  members  of  the  equation. 

The  problem  in  the  second  is  to  raise  a  to  the  a;th  power, 

which  operation  is  called  exponentiation, In  this  operation 

a  is  called  the  base  and  a^  the  exponential  of  x  with  regard 
to  the  base  a. 

In  the  third  case  the  operation  of  finding  x  when  a  and  y 
are  given  is  called  the  log  arithmetic  operation,  and  is  ex- 
pressed in  symbols  by  the  equation  2;=log«?/.  From  this 
it  is  seen  that  the  equation  a^  =  y  may  be  written  in  the 
form  a^^'^'^y  =  y.  The  logarithm  of  a  number  y  to  the  base  "  a  " 
is  that  exponent  which  indicates  the  power  to  tvhich  the  base 
must  be  raised  in  order  to  produce  y.  The  expression  log^  y 
is  read  ''logarithm  of  y  with  respect  to  the  base  a."  It  is 
seen  that  log^?/,  when  found,  is  simply  an  exponent,  and  as 
such  is  subject  to  the  laws  of  indices. 

268.  Since  a^  =  1  for  all  finite  values  of  a  different  from 
zero,  it  follows  that  log^  1  =  0.  Since  a^  =  a,  it  follows  that 
log^a  =  l. 

217 


218  COLLEGE   ALGEBRA 

Since  a^"^  =  go  ,  and  a""^  =  0,  for  all  finite  values  of  a  >  1, 
it    follows    for   such    values    of    a^    that    log^  0  =  —  oo    and 

log«QO   =  +QO. 

Similarly  for  all  positive  values  of  a  <  1,  log„  0  =  oo  and 

log^QO     =-0O. 

If  <^  >  0  and  a;  is  a  real  number,  a^  cannot  be  negative, 
therefore  the  logarithm  of  a  negative  number  is  not  real. 

If  a^  =  m,  a'^  =  7^,  and  m  >  n^  that  is,  if  a^  >  (X^,  it  is  obvious 
that  x> y  Avhen  a >  1,  that  is,  when  the  base  is  greater  than 
unity,  the  greater  the  number  the  greater  the  logarithm, 
and  conversely. 

269.    Theorems.     Let  a^  =  w,  a^  —  n,  then  mn  =  a^a^  =  a-^"^^. 
Therefore,     log^  (w>i)  =  x  +  y  =  log^  m  +  log„  w. 
Thus,  if  logio  2  =  0.3010  and  log^^  3  =  0.4771, 

then  logio  ^  =  ^^^^lo  ^  +  logio  3  =  0.7781. 

Similarly, 

logaCmn  •'•p')=loga(mn"-}-\-log„p 

=  l0gam  -slogan  -\-    '"   +l0gap, 

that  is,  the  logarithm  of  a  product  is  equal  to  the  sum  of  the 
logarithins  of  its  factors. 

Ag-ain,  —  =  —  =  a^~y. 

^  n      ay 

Therefore,  log^  {  —  \=x  —  y  =  log«  m  —  log,,  ?^, 

that  is,  the  logarithm  of  a  quotient  is  equal  to  the  logarithm  of 
the  dividend  minus  the  logarithm  of  the  divisor. 

Thus 
logio  5  =  If^gio  1^  -  l^g'io-  =  1.0000  -  0.3010  =  0.6970. 


LOGARITHMS  219 

Raising  both  sides  of  the  equation  a^  =  m  to  the  hih. 
power,  where  k  is  any  real  number,  integral  or  fractional, 
positive  or  negative,  we  have  a'^^'  =  m^. 

Therefore,  by  definition,  loga  (m*)  =k  •  x=  k log« 771,  that 
is,  the  logarithm  of  any  power  of  a  nurnher  is  the  logarithm  of 
the  number  77mltiplied  hy  the  index  of  the  power ^  whether  the 
index  he  integral  or  fractional^  positive  or  negative. 

Thus  logio  *^  =  logio  2'  =  3  log,,  2  =  0. 9080, 

and  log,„^2  =  logio  2'  =  1  log,„  2  =  0.1008. 

Note.  Since  the  remainder,  when  divided  by  the  divisor,  gives  a 
quotient  which  is  less  than  one  half,  it  is  neglected  ;  if  that  quotient  were 
greater  than  one  half,  it  would  be  called  unitij. 

p 
For  a  base  a  >  1,  log^  ~  =  logap  —  loga  ^,  which  is  positive 

or  negative  according  as  p^qi  that  is,  the  logarithm  of  a 
number  greater  than  unity  is  positive  and  of  a  number  less 
than  unity  is  negative. 

If   1  >  a  >  0   and   a^  >  a^,  then   f  -  J   >  (     )     and  therefore 

y>x^  and  the  greater  the  number  the  less  the  logarithm,  and 

p 
conversely.     The  log„  — =  log„^  —  log^^,   which  is  positive 

or  negative  according  as  p^q.  Therefore  with  this  base 
the  logarithm  of  a  number  greater  than  unity  is  negative 
and  of  a  number  less  then  unity  is  positive. 

We  have  seen,  therefore,  that  for  a  base  greater  than  unity 
the  logarithm  of  a  number  greater  than  unity  is  positive  and  of 
a  number  less  than  uiiity  is  yiegative;  ivhile  for  a  base  less  than 
unity  the  logarithm  of  a  number  greater  than  unity  is  negative 
and  of  a  number  less  than  unity  is  positive.     The  same  result 

may  be  arrived  at  by  substituting  -  for  5  in  the  transforma- 
tion formula  of  271. 


220  COLLEGE   ALGEBRA 

270.  EXAMPLES 

1.    If  a  series  of  numbers  are  in  G.  P.  their  logarithms  are 

in  A.  P. 

2.  Given  2-^  =  8,  find  x. 

3.  Given  logg  27  =  a;,  find  x. 

4.  Given  log^  32  =  5,  find  x. 

Given  log  2  =0.3010,  log  3  =  0.4771,  log  7  =  0.8451;  find 
the  logarithms  of  the  following  numbers  to  the  same  base: 

5.  14.  8.    32.  11.    14f  14.    -^96. 

6.  28.  9.    101.  12.    2.31.  15.     v/48. 

7.  24.  10.    20f  13.    -^'M.  16.    ^I375. 

17.  Find  the  logarithm  of  243  to  the  base  9. 
Let  X  be  the  required  logarithm. 

Then  9-^=243. 

But  9  and  243  are  both  powers  of  3.     Hence  3^^  =  3^, 
^^^  2:r  =  5,  or  2;=  2.5. 

18.  Find  the  logarithm  of  32  to  the  base  4. 

19.  Find  the  logarithm  of  4  to  the  base  32. 

20.  Find  the  logarithm  of  3^3  to  the  base  49. 

21.  Find  the  logarithm  of  343  to  the  base  ^^. 

22.  Find  the  logarithm  of  ^V  to  the  base  |. 

271.  Let  log^  n  =  x^  log^  n  =  y^  then 

a''  =  71  and  h^  =  n 
and  therefore  0^  =  1^. 


LOGARITHMS  221 

Taking  the  logarithm  of  both  members  of  the  last  equation 
with  respect  to  any  third  base  c,  we  have 

or  by  substituting  the  values  of  x  and  ?/, 
log«  '^  log,  a  =  logft  n  log,  b, 

whence  log^  n  =  — ^^^  log^  n. 

Since  o  was  any  real  number  whatevei',  the  ratio  — ^^^^—  is 

constant  for  any  given  value  of  a  and  ^,  and  is  called  the 
moduhis  of  the  tra7isformation. 

T^  1  loo",  a  1 

i^or  c=  a,  we  liave      ^'^    — 


log,  5      log,,^' 

and  log5  n  =  - — —  log„  n. 

loga  0 

This  is  a  formula  for  transforming  logarithms  of  numbers 
which  are  known  with  respect  to  a  base  a  into  logarithms  of 
those  numbers  with  respect  to  any  other  base  b. 

The  student  may  show  that  log,^  b  log^  a  =  l. 

272.  Although  theoretically  any  positive  number  except 
unity  could  be  made  the  base  of  a  S3^stem  of  logarithms,  yet 
for  practical  purposes  only  two  systems  are  at  all  frequently 
used.  One,  called  the  natural  system^  or  sometimes  the 
Napierian  system^  is  explained  in  a  subsequent  article,  283. 
The  other,  known  as  the  Briggs^  or  common^  system^  has  the 
number  10  for  its  base.  It  is  used  for  all  purposes  involving 
merely  numerical  calculation.  The  advantage  of  this  base 
consists  in  the  fact  that  any  change  in  the  position  of  the 
decimal  j)oint  in  a  number  will  merely  add  an  integer  to,  or 
subtract  it  from,  the  logarithm,  because  the  number  will  then 


222  COLLEGE   ALGEBRA 

merely  be  multiplied  or  divided  by  an  integral  power  of  10, 
269.  Hence  the  fractional  portion  of  the  logarithm  will  be 
the  same  for  any  sequence  of  figures,  whatever  the  position 
of  the  decimal  point  among  them. 

273.  In  the  Briggs  system  the  logarithm  of  1  is  0,  and  the 
logarithm  of  10  is  1,  by  268.  Hence  the  logarithm  of  any 
number  between  1  and  10  is  a  positive  fraction.  Every 
number  greater  than  10  may  be  obtained  by  multiplying  a 
number  between  1  and  10  by  an  integral  power  of  10.  Hence 
its  logarithm  consists  of  an  integer  plus  a  fraction.  This 
integer,  which  has  been  shown  to  be  dependent  merely  upon 
the  position  of  the  decimal  point  in  the  number,  is  called  the 
characteristic  of  the  logarithm.  The  fractional  part,  which 
depends  merely  upon  the  sequence  of  figures  in  the  number, 
and  is  ordinarily  written  in  the  form  of  a  decimal,  is  called 
the  mantissa  of  the  loga7'ithn.  The  mantissa  is  ordinarily  not  a 
terminating  decimal,  but  is  carried  out  four,  five,  six,  etc.  places 
according  to  the  degree  of  accuracy  required  in  the  work. 

Thus  log  2  =  0.3010,  and  log  200  =  2.3010.  In  each  case 
the  mantissa  is  .3010,  while  the  characteristics  are,  respec- 
tively, 0  and  2. 

274.  The  logarithm  of  a  positive  number  less  than  unity 
is  really  negative,  but  since  such  a  number  may  be  derived 
from  a  number  between  1  and  10  by  dividing  it  by  an  in- 
tegral power  of  10,  it  is  convenient  to  regard  the  logarithm 
as  composed  of  a  positive  mantissa  and  a  negative  charac- 
teristic. Thus  if  log  2  =  0.3010,  then  since  .002  =  -?-, 
log  .002  =  0.3010-3. 

275.  Two  methods  are  in  common  use  for  writing  loga- 
rithms with  negative  characteristics.  Thus  log  .002  =  3.3010, 
where  the  negative  sign  is  placed  over  the  characteristic  to 
indicate  that  it  alone  is  negative ;  or  the  —  3  is  called  7  —  10 


LOGARITHMS  223 

and  tlie  logarithm  is  written  7.3010  —  10.  By  this  means 
the  negative  portion  is  always  a  multiple  of  10,  and  is  kept 
quite  separate  from  the  positive  portion  of  the  logarithm. 

276.    To  find  the  logarithm  of  a  given  number. 

1.  To  find  the  characteristic.  In  273  we  have  seen  that  if 
the  decimal  point  follows  the  first  significant  digit,  the  charac- 
teristic of  the  logarithm  is  0.  For  every  place  the  decimal 
point  in  the  number  is  moved  toward  the  right  the  number  is 
multiplied  by  10.  Hence  if  the  number  is  greater  than  10, 
the  characteristic  is  one  less  than  the  number  of  significant 
figures  to  the  left  of  the  decimal  point.  Likewise  for  every 
place  the  decimal  point  is  moved  toward  the  left  the  number 
is  divided  by  10.  Hence  if  the  decimal  point  immediately 
precedes  the  first  significant  digit,  the  characteristic  of  the 
logarithm  is  —  1,  or  9  —  10.  If  one  cipher  intervenes,  it  is 
8  —  10,  and  so  on,  subtracting  one  for  each  additional  cipher. 

The  characteristic  should  be  written  first,  and  always  ex- 
pressed even  though  it  be  zero,  in  order  to  avoid  error  due  to 
forgetting  it. 

2.  To  find  the  mantissa  from  the  table,  (a)  When  the 
number  has  just  three  significant  figures.  Pages  224  and  225 
give  a  table  of  the  mantissoe  of  the  Briggs  logarithms  of  all 
integers  from  1  to  1000.  In  order  to  find  the  mantissa  of  a 
given  number,  look  for  the  first  two  digits  in  the  column 
marked  N.  These  indicate  the  row  in  Avhich  the  mantissa 
is  to  be  found.  The  column  is  designated  by  the  third  digit. 
Thus  the  mantissa  for  478  is  6794,  and  the  entire  logarithm 
is  2.6794  by  1. 

(5)  When  the  number  has  less  than  three  significant 
dibits.  To  find  the  log^arithm  of  .7  we  look  for  700  in  the 
tables  and  find  the  mantissa  8451.  Hence  log  .7  =  1.8451, 
or  9.8451-10. 


224 


COLLEGE   ALGEBRA 


N 

0 

1 

2 

3 

4      5 

6 

7 

8 

9 

0 

0000 

0000 

3010 

4771 

6021 

6990 

7782 

8451 

9031 

9542 

1 

0000 

0414 

0792 

1139 

1461 

1761 

2041 

2304 

2553 

2788 

2 

3010 

3222 

3424 

3617 

3802 

3979 

4150 

4314 

4472 

4624 

3 

4771 

4914 

5051 

5185 

5315 

5441 

5563 

5682 

5798 

5911 

4 

6021 

6128 

6232 

6335 

6435 

6532 

6628 

6721 

6812 

6902 

5 

6990 

7076 

7160 

7243 

7324 

7404 

7482 

7559 

7634 

7709 

6 

7782 

7853 

7924 

7993 

8062 

8129 

8195 

8261 

8325 

8388 

7 

8451 

85i3 

8573 

8633 

8692 

8751 

8808 

8865 

8921 

8976 

8 

9031 

9085 

9138 

9191 

9243 

9294 

9345 

9395 

9445 

9494 

9 
10 

9542 

9590 

9638 

9685 

9731 

9777 

9823 

9868 

9912 

9956 

0000 

0043 

0086 

0128 

0170 

0212 

0253 

0294 

0334 

0374 

11 

0414 

0453 

0492 

0531 

0569 

0607 

0645 

0682 

0719 

0755 

12 

0792 

0828 

0864 

0899 

0934 

0969 

1004 

1038 

1072 

1106 

13 

1139 

1173 

1206 

1239 

1271 

1303 

1335 

1367 

1399 

1430 

14 

1461 

1492 

1523 

1553 

1584 

1614 

1644 

1673 

1703 

1732 

15 

1761 

1790 

1818 

1847 

1875 

1903 

1931 

1959 

1987 

2014 

16 

2041 

20(J8 

2095 

2122 

2148 

2175 

2201 

2227 

2253 

2279 

17 

2304 

2330 

2355 

2380 

2405 

2430 

2455 

2480 

2504 

2529 

18 

2553 

2577 

2601 

2625 

2648 

2672 

2695 

2718 

2742 

2765 

19 
20 

2788 

2810 

2833 

2856 

2878 

2900 

2923 

2945 

2967 

2989 

3010 

3032 

3064 

3075 

3096 

3138 

3139 

3160 

3181 

3201 

21 

3222 

3243 

3263 

3284 

3304 

3324 

3345 

3365 

3385 

3404 

22 

3424 

3444 

3464 

3483 

3502 

3522 

3541 

3560 

3579 

3598 

23 

3617 

3636 

3655 

3674 

3692 

3711 

3729 

3747 

3766 

3784 

24 

3802 

3820 

3838 

3856 

3874 

3892 

3909 

3927 

3945 

3962 

25 

3979 

3997 

4014 

4031 

4048 

4065 

4082 

4099 

4116 

4133 

26 

4150 

4166 

4183 

4200 

4216 

4232 

4249 

4265 

4281 

4298 

27 

4314 

4330 

4346 

4362 

4378 

4393 

4409 

4425 

4440 

4456 

28 

4472 

4487 

4502 

4518 

4533 

4548 

4564 

4579 

4594 

4()09 

29 
30 

4624 

4639 

4654 

4669 

4683 

4698 

4713 

4728 

4742 

4757 

4771 

4786 

4800 

4814 

4829 

4843 

4857 

4871 

4886 

4900 

31 

4914 

4928 

4942 

4955 

4969 

4983 

4997 

5011 

5024 

5038 

32 

5051 

5065 

5079 

5092 

5105 

5119 

5132 

5145 

5159 

5172 

33 

5185 

5198 

5211 

5224 

5237 

5250 

5263 

5276 

5289 

5302 

34 

5315 

5328 

5340 

5353 

5366 

5378 

5391 

5403 

5416 

5428 

35 

5441 

5453 

5465 

5478 

5490 

5502 

5514 

5527 

5539 

5551 

36 

5563 

5575 

5587 

5599 

5611 

5623 

5635 

5647 

5658 

5670 

37 

5682 

5694 

5705 

5717 

5729 

5740 

5752 

5763 

5775 

5786 

38 

5798 

5809 

5821 

5832 

5843 

5855 

5866 

5877 

5888 

5899 

39 
40 

5911 

5922 

5933 

5944 

5955 

5966 

5977 

5988 

5999 

6010 

6021 

6031 

6042 

6053 

6064 

6075 

6085 

6096 

6107 

()117 

41 

6128 

6138 

6149 

6160 

6170 

6180 

6191 

6201 

6212 

6222 

42 

6232 

6243 

6253 

6263 

6274 

6284 

6294 

6304 

6314 

6325 

43 

6335 

6345 

6355 

6365 

6375 

6385 

6395 

6405 

6415 

6425 

44 

6435 

6444 

6454 

6464 

6474 

6484 

6493 

6503 

6513 

6522 

45 

6532 

6542 

6551 

6561 

6571 

6580 

6590 

6599 

6609 

6618 

46 

6628 

6(i37 

664(5 

()()56 

6665 

6675 

6684 

6693 

6702 

6712 

47 

6721 

6730 

6739 

6749 

6758 

6767 

6776 

6785 

671H 

6803 

48 

6812 

6821 

6830 

6839 

6848 

6857 

68()6 

6875 

6884 

(;S!)3 

49 

6^X)2 

6911 

6920 

6928 

6937 

6946 

6955 

6964 

6972 

6981 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

LOGAKITIIMS 


225 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

60 

6990 

6998 

7007 

7016 

7024 

7033 

7042 

7050 

7059 

7067 

51 

7076 

7084 

7093 

7101 

7110 

7118 

7126 

7135 

7143 

7152 

52 

7160 

7168 

7177 

7185 

7193 

7202 

7210 

7218 

7226 

7235 

53 

7243 

7251 

7259 

7267 

7275 

7284 

7292 

7300 

7:308 

7316 

54 

7324 

7332 

7340 

7348 

7356 

7364 

7372 

7380 

7388 

7396 

55 

7404 

7412 

7419 

7427 

7435 

7443 

7451 

7459 

7466 

7474 

5() 

7482 

74W 

7497 

7505 

7513 

7520 

7528 

15m 

7543 

7551 

57 

75.")!) 

7566 

7574 

7582 

7589 

7597 

7604 

7612 

7619 

7(527 

58 

7634 

7(542 

7649 

7657 

7664 

7672 

7679 

7686 

7694 

7701 

59 
60 

7709 

7716 

7723 

7731 

7738 

7745 

7752 

7760 

7767 

7774 

7782 

7789 

779(5 

7803 

7810 

7818 

7825 

7832 

7839 

7846 

(31 

7853 

7860 

7868 

7875 

7882 

7889 

7896 

7903 

7910 

7917 

62 

7924 

7931 

7938 

7945 

7952 

7959 

7J366 

7973 

7980 

7987 

03 

7993 

8000 

8007 

8014 

8021 

8028 

8035 

8041 

8048 

8055 

64 

8062 

8069 

8075 

8082 

8089 

8096 

8102 

8109 

8116 

8122 

65 

8129 

8136 

8142 

8149 

8156 

8162 

8169 

8176 

8182 

8189 

66 

8195 

8202 

8209 

8215 

8222 

8228 

8235 

8241 

8248 

8254 

67 

8261 

8267 

8274 

8280 

8287 

8293 

8299 

8;306 

8312 

8319 

68 

8325 

8331 

8338 

8344 

8351 

8357 

8363 

8370 

8376 

8382 

69 
70 

8388 

8395 

8401 

8407 

8414 

8420 

8426 

8432 

8439 

8445 

8451 

8457 

8463 

8470 

8476 

8482 

8488 

8494 

8500 

8506 

71 

8513 

8519 

8525 

8531 

8537 

8543 

8549 

8555 

8561 

8567 

72 

8573 

8579 

8585 

8591 

8597 

8603 

8609 

8615 

8(321 

8627 

73 

8633 

8639 

8645 

8651 

8657 

8663 

8(5(39 

8675 

8681 

8(38(5 

74 

8692 

8698 

8704 

8710 

8716 

8722 

8727 

8733 

8739 

8745 

75 

8751 

8756 

8762 

8768 

8774 

8779 

8785 

8791 

8797 

8802 

76 

8808 

8814 

8820 

8825 

8831 

8837 

8842 

8848 

8854 

8859 

77 

8865 

8871 

8876 

8882 

8887 

8893 

8899 

8904 

8910 

8915 

78 

8921 

8927 

8932 

8938 

8943 

8949 

8954 

8960 

8965 

8971 

79 
80 

8976 

8982 

8987 

8993 

8998 

9004 

9009 

9015 

9020 

9025 

9031 

9036 

9042 

9047 

9053 

9058 

9063 

9069 

9074 

9079 

81 

9085 

9090 

9096 

9101 

9106 

9112 

9117 

9122 

9128 

9133 

82 

9138 

9143 

9149 

9154 

9159 

9165 

9170 

9175 

9180 

9186 

83 

9191 

9196 

9201 

9206 

9212 

9217 

9222 

9227 

9232 

9238 

84 

9243 

9248 

9253 

9258 

9263 

9269 

9274 

9279 

9284 

9289 

85 

9294 

9299 

9304 

9309 

9315 

9320 

9325 

9330 

9335 

9340 

86 

9345 

9350 

9355 

9360 

9365 

9370 

9375 

9380 

9385 

93f)0 

87 

9395 

9400 

9405 

9410 

9415 

9420 

9425 

94130 

9435 

f)440 

88 

9445 

9450 

9455 

94(50 

9465 

9469 

9474 

t|479 

9484 

9489 

89 
90 

9494 

9499 

9504 

9509 

9513 

9518 

9523 

9528 

9533 

9538 

9542 

9547 

9552 

9557 

9562 

9566 

9571 

9576 

9581 

9586 

91 

9590 

9595 

9600 

i)(505 

9609 

9614 

9619 

9624 

9628 

9633 

92 

9638 

9(543 

9(547 

m52 

9657 

9661 

9666 

mn 

9675 

9680 

93 

9685 

9(589 

9694 

9699 

9703 

9708 

9713 

9717 

9722 

9727 

94 

9731 

9736 

9741 

9745 

9750 

9754 

9759 

9763 

9768 

9773 

95 

9777 

9782 

9786 

9791 

9795 

9800 

9805 

9809 

9814 

9818 

96 

9823 

9827 

9832 

9836 

9841 

9M5 

9850 

9854 

9859 

9863 

97 

98(i8 

9872 

9877 

9881 

9886 

9890 

9894 

9899 

9<X)3 

9908 

98 

9912 

9917 

9921 

9926 

9930 

99:i4 

9939 

9943 

9948 

9952 

99 

9956 

9961 

9965 

99(59 

9974 

9978 

9983 

9987 

9991 

9996 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

9f. 


226  COLLEGE   ALGEBRA 

(6?)  When  the  number  has  more  than  three  significant 
digits. 

Thus  in  the  case  of  log  32.456,  since  32.456  lies  .5Q  of  the 
way  from  32.4  to  32.5,  its  logarithm  must  lie  about  .56  of 
the  way  from  log  32.4  to  log  32.5.  But  log  32.4  =  1.5105 
and  log  32.5  =  1.5119.  Hence,  leaving  the  decimal  point 
out  of  account,  the  increase,  or  tabular  difference  as  it  is 
called,  is  14,  and  .36  of  this  is  7.84.  Hence,  adding  this  cor- 
rection to  log  32.4,  we  have  log  32.456  =  1.5113.  Since  our 
tables  are  given  only  to  four  decimal  places,  we  retain  only 
four  in  correction,  always  figuring  the  fourth  place  to  the 
nearest  unit,  thus  in  this  case  adding  8  as  the  correction. 

Likewise  for  log  .0035678,  we  find  from  the  tables  log 
.00356  =  7.5514  -  10,  and  the  tabular  difference  is  13.  The 
correction  (13  x  .78)  is  10,  so  that  log  .0035678  =  7.5524-10. 

Find  the  logarithms  of  the  following  numbers : 

1.  428.  5.  .524.  9.  .050009. 

2.  327.  6.  .02345.  10.  .0042085. 

3.  82.46.  7.  .38634.  ii.  20.308. 

4.  32.875.  8.  430.23.  12.  7,352,000. 

277.  To  find  the  number  corresponding  to  a  given  logarithm. 

Given  log  x=  3.3765 ;   to  find  x. 

This  is  the  reverse  operation  of  that  given  in  276.  Since 
the  characteristic  merely  determines  the  position  of  the 
decimal  point,  273,  we  look  for  the  mantissa  in  the  table. 
The  next  smaller  mantissa  in  the  table  is  3747,  which  corre- 
sponds to  the  number  237.  The  excess  of  3765  over  3747  is 
18.  The  tabular  difference  as  found  from  the  table  by  sub- 
tracting 3747  from  3766  is  19.  Hence  3765  is  If  of  the  way 
from  3747  to  3766,  and  the  corresponding  number  is  about 
i|  of  the  way  from  237  to  238,  or,  reducing  IJ  to  a  decimal. 


LOGARITHMS  227 

about  .95  of  a  unit  beyond  237.  Hence  the  corresponding 
digits  are  23795,  and  since  the  characteristic  is  3,  the  deci- 
mal point  must  be  moved  three  places  to  the  right  of  where 
it  would  be  placed  for  a  zero  characteristic,  and  hence 
a;  =  2379.5. 

Similarly,  if  log  ^  =  7.3765-10,  ?/  =  . 0023795,  since  in 
this  case  the  characteristic  is  7  —  10,  or  —  3,  and  the  decimal 
point  must  be  moved  three  places  to  the  left. 

Find  the  numbers  corresponding  to  the  following  loga- 
rithms : 


1. 

2.8987. 

7. 

4.6062. 

13. 

5.8124. 

2. 

3.5705. 

8. 

2.6842. 

14. 

0.7318. 

3. 

0.7016. 

9. 

1.3427. 

15. 

4.7306-10. 

4. 

9.8814- 

-10. 

10. 

0.4850. 

16. 

5.4783. 

5. 

1.9542. 

11. 

7.6123. 

17. 

2.7005. 

6. 

3.6558. 

12. 

8.5493- 

-10. 

18. 

1.6100. 

278.    Cologarithms.     The  cologarithm  of  a  number   is   the 
logarithm  of  the  reciprocal  of  the  number. 

Thus  colog  425  =  log  -^^  =  log  1  -  log  425  (269) 

425 

=  0  -  2.6284. 

But  since  we  always  wish  to  have  the  mantissa  of  a  loga- 
rithm positive,  we  write  0=10—10,  and  subtract  2.6284 
from  this,  as  follows: 

log  1  =  10.0000  -  10 
log  425=    2.6284 


colog  425=    7.3716-10. 


228  COLLEGE   ALGEBRA 

In  practice  this  is  done  mentally  by  beginning  at  the  left 
and  subtracting  each  digit  from  9,  except  the  last  significant 
digit,  which  is  subtracted  from  10. 

279.    Computation  by  logarithms. 

EXAMPLES 

1.  Find  the  value  of  2345  x  2.327  x  .004296. 

log  (2345  X  2.327  x  .004296)  =  log  2345  +  log  2.327 

+  log  .004296.        (269) 
log  2345  =  3.3702 
log  2.327  =  0.3668 

log.004296  =  7.6331 -10 

log  (2345  X  2.327  x  .004296)  =  1.3701.        ' 

3692 
19)9.00 
.47 
Therefore,  2345  x  2.327  x  .004296  =  23.447. 

Note.  It  must  be  borne  in  mind  that  these  logarithms  are  only  approxi- 
mations carried  out  to  a  number  of  decimal  places  determined  by  the  number 
of  decimal  places  of  the  table  used.  Four-place  tables  do  not  give  even  five 
figures  in  the  result  with  any  considerable  degree  of  accuracy.  If  greater 
accuracy  is  desired,  a  table  with  more  decimal  places  must  be  used. 

2.  Find  the  value  of '-^ '—^ — -• 

.002578  X  386.5 

Since  this  is  the  product  of  two  numbers  divided  by  the 
product  of  two  others,  it  might  be  solved  by  subtracting  the 
sum  of  the  logarithms  of  the  factors  of  the  denominator  from 
the  sum  of  the  logarithms  of  the  factors  of  the  numerator. 
But  this  would  require  several  operations,  and  it  is  customary 
to  regard  the  fraction  as  the  continued  product  of  the  factors 
of  the  numerator  and  the  reciprocals  of  the  factors  of  the 
denominator. 


LOGARITHMS  229 

Thus 

37.54  X  .02486  ^        /g^^^^  ^  ^^2436  x  -^  x  -^.^ 
•  ^  .002578  X  886.5         "^  V  .002578      386.5; 

=  log  37.54  +  log  .02436  +  colog  .002578 

+  colog  386.5. 
log  37.54  =  1.5745 

log  .02486=8.3867-10 

colog  .002578  =  2.5887 

colog  386.5  =  7.4128 -10 

^87.54  X. 02486  ^^g^,^_^Q^ 
^  .002578  X  386.5 

Hence  87.54  x  .02436  ^   ,^,,3^ 

.002578  X  386.5 

3.    Find  the  value  of  (38.64)6. 

log  (38.64)6  =  6  X  log  38.64.  (269) 

log  38.64  =  1.5870. 
Therefore  log  (38. 64)6  ^  9. 5220, 

and  (38.64)6  =  3,326,900,000. 


4.    Find  the  value  of  V7f684. 


log  V.7684  =  ilog  .7684. 
log  .7684  =  9.8856-10 
=  49.8856-50. 
Therefore  log  </7mi  =  9.9771  -  10, 


J' 

5j 


and  V.  7684  =  .9486 


230  COLLEGE   ALGEBRA 

Find  by  means  of  logarithms  the  approximate  values  of 
the  following  expressions : 

6.    234  X  345  X  456.  ^     48. 8T  x  .03245  x  389.2 


6.    2345  X  3456  x  4567. 


10. 


7.459  X  9.351 
.07371  X  907.3  x  .6007 


7.    86.23  X  23.45  X. 08632.        ^^'  .7623x8.076 

28.42  X  67.54x96.72         ^^      -7.831x3.867x8.903 
92.57  X  13.83         *  "  .007459  x  12.87 

Note.  Since  all  real  powers  of  10  are  positive,  negative  numbers  have 
no  real  logarithms.  Hence  examples  involving  negative  numbers  must  be 
worked  as  if  the  numbers  were  positive,  and  then  the  proper  sign  is  to  be 
attached  to  result. 

^2     r-. 008734)  X  (-  8.345)  x  834.7 
(-.8793)  X  (-900.6) 

13.    (8.341)*.  19.    (-.5429)1 

'''    ^''''''^''  20.    (-.05387)1. 

'''  (^^•^^);-  21.  (.3841)1 

16.  (56.38)^.  ^^  V3596  x  ^:4287 

17.  (.02583)i  a/. 0586 

18.  (-8.425)1  23.  aJ-V:5804  x  V^MO^. 

24.  Find  by  means  of  logarithms  the  amount  of  f  486  in 
five  years  at  five  per  cent  if  the  interest  is  compounded 
annually. 

25.  Find  the  amount  of  -1384  in  forty  years  at  four  per 
cent  if  the  interest  is  compounded  semiannually. 

26.  Solve  the  equation  4^  =  246. 


LOGARITHMS  231 

Taking  the  logarithm  of  each  member,  we  have 

X  log  4  =  log  246, 

lo^  246      2.3909      .  o-i 

or  x  =  -^ ■  = T  =  3.9<1. 

log  4        0.6021 

27.  Solve  the  equation  415^^^^  =  SIT'''^"^ 

28.  Find  the  logarithm  of  428  to  the  base  .8. 

If  X  represent  the  required  logarithm, 

.8^^=428, 

,  los^  428  ^^  .,1     otri\ 

and  X  =  — (Compare  with  271) 

log  .8 

2.6314  2.6314  ^r. -.^ 

=  —  z< .lb. 


9.9031-10  0.0969 

Note.  In  this  case  9.9031  —  10  is  really  a  binomial  expression  and  hence 
must  be  combined  in  a  single  monomial  before  ordinary  arithmetical  divi- 
sion can  be  performed. 

29.  Find  the  logarithm  of  376  to  the  base  12. 

30.  Find  the  logarithm  of  .536  to  the  base  7. 

thp:  exponential  function 

280.  Definition.     Let  us  deiine  F(x^  as  the  limit  of  ll  -\ —  )  , 
as  w  =  CO,  that  is,  -F(a:)  =  L  f  1  +  -  j  ,  for  both  real  and  complex 

7  /•  7  71=30   V  nJ 

values  oj  x  ana  n. 

281.  From  the  foregoing  definition  we  easily  derive  the 
following  properties  of  FQx~). 

1.   When  x=0,  we  have  ^(0)  =  L(iy  =  1.* 

*  Although  I*'  must  be  considered  an  indeterminate  form  when  it  is  the 
limit  of  a  variable  which  approaches  1  raised  to  an  infinite  power,  here  we 
have  strictly  a  constant. 


232  COLLEGE   ALGEBRA 

2.  For  all  real  values  of  x  and  n^  F(x)  is  the  same  for  a 
given  value  of  x  in  whatever  way  n  becomes  indefinitely 
great.  For  if  n  is  any  real  positive  fraction,  we  can  always 
choose  two  consecutive  positive  integers  such  that 

Hence,  according  as  x  is  positive  or  negative, 

X     <x<:^  X 


m 


+  1  -^  'nr  m 


and  l4.^>l  +  ^>l+_^, 

and  hence     (l  +  -T^^  >  f  1  +  -  Y'  >(l  + 
\        mj  \        nj       \ 


1  + 


m  + 1, 


V     m  +  iy 

Taking   the  limits  as  n  =  oo,   and  hence  as  w  =  go,  and 
771  -j-  1  =  00,  and  denoting  by  /(a;)  the  value  of  the  limit  of 

(l-\ — )     as  m,  while  remaining  a  positive  integer,  becomes 
\        'rnj  ,  ^ 

indefinitely  great,  we  see  that  ( 1  +  -  j  ,  lying  between  two 

numbers  each  of  which  approaches  the  limit /(a:;),  must  itself 
approach /(rr)  as  its  limit,  or 

when  n  is  positive,  whether  x  is  positive  or  negative. 

*  For  x  <  0,  the  inequality  sign  and  the  order  of  the  exponents  would  be 
reversed. 


LOGARITHMS  233 

li  n  =  —  p^  where  jt?  is  a  positive  number,  then 


n=oo\  ^/  p=ao\  pJ  p^coXp        X. 

p-x^^\  p  —  xj         \  p  —  Xj 

We  have  proved  therefore  that 
for  all  real  values  of  x  and  n. 


If  :?:  =  1,  we  have    ifl  +  iY  =  /(!). 


3.  Since  X  fl  +  -Y'  =  l{[\^-  ^  p Y , 
we  have,  putting  -  =  m, 

or  f(x)  =  /(1)%  for  all  real  values  of  x. 
In  particular,  /(  —a;)  =/(l)"^. 

4.  By  (3),  we  have 

or  /(^)/(i/)=/(^  +  i/). 

The  second  property  shows  that  for  all  real  values  of  x 
^'''^^'  Fix)=f(ix^. 


234  COLLEGE   ALGEBRA 

282.    Expanding  f  1  -f  -  J  by  the  binomial  theorem,  we  have 

1  +  -     =l+-x+  ^  — +  •- 

\        nJ  n  z  I        n^ 


1-1 


+  -^ -. --+  -"  =1+2;  +  — —, —  x^  + 

r\  7f  ^  I 


i_iYi_2v.Yi_!i^ 


+ ^ ^^  +  ••••   (1) 

Taking  the  limits  of  both  sides  as  w  =  go,  we  have 

That  the  right-hand  member  has  this  limit  may  be  seen  as 
follows  : 

The  (r  +  l)th  term  is 

i_iYi-2V..ri-':^ 


.                    y^               n/\              71/             \                    71       y       r 
tr+i  = X  . 

r  I 
Let  fl^j,  «2»  ^3'  '"•)  ^^6  positive  proper  fractions.     Then,  since 

(1  —  «i)(l  —  ^2)  =  1  —  <^i  —  «2  +  ^1^2' 

1  >  (1  —  «i)  (1  —  ^2)  >  1  —  «i  —  «2- 
Similarly 

1  >  (1  —  <^i)(l  —  (^>2)0^  ~  ^3)  >  1  —  ^1  —  «2  ~~  ^3' 

l>(l-a{)(l-a^)  •••  (1--%.)>1  -aj-«2-^3  "•  -  ^k- 

(2) 


LOGARITHMS  235 

1  2  r  —  1 

Choosinsc  A:  =  r  —  1,  a,  =  -,  «„  =  -,  •••,  cik  = ->  ^^^  multi- 

r  n  n  n 

plying  (2)  by  — -,  we  have,  using  a  positive  x  for  convenience, 

^1 


x^  \ti      n  n 

^  <"  f       > ^  -> 


l_,l  +  2^,..+i^ 


^^  ^\  >  ^r+l  >Z1~ 


x^      ,  x""      J  rCr  —  1)   ^ 

r !       ^^^      r\  n  r\ 


X 


X       .  x^  x^ 

-r>^r+i>-r- 


r\       '"-^      r\      2n  (r-2)! 

Giving  r  the  values  0,  2,  4,  •••,  and  the  values  1,  3,  5,  •••, 
we  get  the  following  two  systems  of  relations  respectively, 

JL  —  Ci  —  X,  X  —  Cq  —  •*/, 

/yi^  /yi^  /yt^  /y^  /yn>  nriO 

2!^  ^^2!~27i'  3T^  ^^3T~2^'  ^^ 


Adding  the  corresponding  terms  of  the  relations  of  each 
system,  we  have 

x^    .    X^    ,  .     ,     .    .     .    ^     .  ^    -I    .    x^    ,    x^ 


2!      4! 


>  ^1  +  ^3  +  ^5+  •••>^  +  2:  +  47+  - 


2  TiV        214! 
-2^1" +3! +  61+- 


236  COLLEGE   ALGEBRA 


As  7i  =  oo,  the  series    1  +  ^  +  —  4-  ••• 


and  ^  +  3T  +  It  + 


are  absolutely  convergent,  212,  16,  the  terms 

27iV        2!      4!  J 


and  f^f^+^  +  ^  + 

2  7iV        3!      5! 


approach  zero,  and  hence  we  see  that 


W=00  O    i  O    I 

Adding  and   subtracting   these  limits  and   remembering 
that 


^l  +  ^2  +  -  +  ^n+l=      I4--      , 


X 


^l-^2+-  +  (-lr^«+l=      1--  h 


n. 


X 


n 


71 


wehave  l  (1  +  ^^  =1 +.  + f,  +  ^^  +  ... ,  (213} 

n^^\nj  213! 


xv      ^  .    x^       x^ 


^         7iJ  2!      3! 


LOGARITHMS  .  237 

This  completes  the  proof  that  for  any  real  values  of  x  and  n^ 

xY      -,    .       ,   x^   .   x^ 


If  in  the  preceding  proof  we  had  taken  x  negative,  the 
right-hand  column  of  relations  (3)  would  have  had  their 
signs  all  reversed,  but  in  taking  the  limits  we  would  have 
arrived  at  the  same  result.* 

283.    The  series 

is  denoted  by  e,  which  is  the  base  of  the  natural  system. 

That  the  quantity  e  is  finite  and  lies  between  2  and  3  may 
be  easily  seen,  for 

6=1  +  1  +  1  + 

_  2+ 

11  11 

and  since         ^  '^  Y\'^  T\^  '"  ^^  ^  2^  ^^'^  '**' 

,         1 

or  e  —  1< 


^         2 

or  e  —  1  <  2. 

Therefore  ^  <  3,  hence  3  >  e  >  2. 

Since /(a:)  =/(l)-^,  we  have  proved  that 

Zt  I       6  1 
for  all  real  values  of  x. 


-n  )  -1 


*  Or  Otherwise         ^  fi-^V^   L  \fi^_±_\    "1 

fix)   jo^y 


238  •         COLLEGE   ALGEBRA 

284.    The  determination  of  the  value  of  F(x)  when  x  and 
n  are  complex  numbers  can  be  made  as  follows : 

If  2  is  a  complex  number  and  n  a  positive  integer,  the  (r  + 1)  th 

term  in  f  1  +  -  j  ^  z  being  equal  to  a;  (cos  (j)  +  i  sin  ^),  is,  380, 


T     = 


[l jfl  —  -j---fl ja:^(cos  r(l)-{-i  sin  r(f)) 


1 )(1  —  -j--.(l ]x^  cos  rcf) 

Liet        t  ^+j  —  —^  , 

1  _  IVl  _  ?y .  Yl  _  r^Vr  sin  r<^ 
nj\        nJ      \  n    J 


and   ^'',.+1  = 


'•+1  —  ^  J 

then  as  before        1  =  ^'.^  =  1, 

X  cos  (^  =  t' ^  =  X  cos  <^, 

-cos2^>^'3>— cos2(/, ^— ^, 

^^    o  /   4/   ^^    o  ji   ^  cos  3  (f) 
— -  cos  3  (/)  >  ^'4  >  —  cos  3  (/) ^, 


X^ 


5 


,  ,   .,   a;*    1  JL   ^*  cos  4  6 
-cos5(^>t'6>— cos5(^-^— ^, 


x^  ±  ^  .1       -^  x^  ,         x"^  COS  n  6 

nl  71 !        2  n  (n  —  2) ! 


LOGAIUTHMS  239 

On  adding,  we  get 

l+a;cos<^  +  — -cos  2<j)-\ \-—-cos7i(t)>t'.-\-t'-\ h^Wi 

x^ 
>  1  H-  a;  cos  <^  +  •  •  •  H r  cos  n^ 

n\ 

x^  f  x^~^  \ 

cos  2  6  +  X  cos  3  (^  +  •  •  •  H ——  cos  ncj) ) . 

2n\  (»i  —  2) !  / 

As  n  =  00,  the  series 

1  -\-x  cos  6  +  •  •  •  H r  cos  n<j> 

n ! 

and  cos  2^  -\-  x  cos  3  <^  H-  •  •  •  H ——cos  w<^ 

are  absolutely  convergent,  since  the  terms  of  each  are  less 
than  the  corresponding  terms  of  the  absolutely  convergent 
series  o  „ 

z !  n\ 

which  as  7^  =  GO   has  the  limit /(a:)  or  e^.     Therefore 

2 

L  (^\  +  ^'2+'-'  +  ^^+i)  =  l  +  -^'co3<^  +  |-cos2<^+....    (1) 

In  the  same  way  it  can  be  proved  that 
L  (^''2  +  ^''3+-+^Vi)=^sin</>  +  ^sin2(^+--.        (2) 

Multiplying  (2)  by  i  and  adding  the  result  to  (1)  and  observ- 
ing that 

V  1  ^^=^  -^  1 '    ^9  "t"  ^  ^    2  ~~       9'    ^  "I  3  ~~       3'    etc., 

and  that  (  T^  +  T^  +  •  •  •  +  ^+1)  =  (l  +  |T ,        (213,  214) 


240  COLLEGE  ALGEBRA 

we  liave 

L  (T,-hT^+'-  +  T^^O  =  L  fl  +  -Y  =  l  +  2;(cos(/,  +  zsin(^) 

71=00  n=^au   \  fl/ 

+  — (cos2(^  +  ^sin2(/))  H , 

The  extension  to  any  real  value  of  n  may  be  made  by 
considering    the    limits    of    the   moduli    and    arguments    of 

1  -\ ,1+-,1h —  and  finally  extending  to 

m  +  ly      \        nj     \        mj 

a  negative  n. 

Hence,  when  z  is  complex  and  n  is  real, 

If  both  z  and  n  are  complex,  we  have,  if 
n  =  m  (cos  (f)  -{-i  sin  0), 

X    1  +  ^1  =£  (1+  ^ 


7i/       7n=oo  v        m  (cos  (^  +  *'  sin  (^^ 

—  L    f(l-\-  ^(CQS  (^  -  ^  sin  (f)~^yn\  cos <t,+i sin  ^ 
m=x>  W  m  J    J 

=  (/(2(cos(^-2sin<^)))^««'^  +  ^'«^"*^. 

This  is  as  far  as  we  can  carry  the  proof.  If,  however,  we 
agree  to  give  to  a  complex  exponent  such  an  interpretation 
that  the  third  property,  viz.  f(z)  =  (/(l))^,  shall  still  hold 
even  when  z  is  complex,  we  have 

F{Z^  =  (/(^(COS  (^  -  Z  sin  <^)))cos«  +  ism<^ 


LOGARITHiMS  241 

Thus,  for  all  values  of  x  and  71,  we  have 

Fix)  =f{x). 

That  the  series  denoted  by  f{x)  is  convergent  has  been 
seen  from  the  mode  of  its  derivation,  since  each  of  the  con- 
stituent series  of  which  it  is  composed  is  convergent  whether 
X  be  real  or  complex.     The  result 

1  +  2)   =,-  =  !+:,+ £-  +  |.^  +  ... 

is  known  as  the  Exponential  Theorem, 
285.    Since  a''=  e^^o^e",  we  have 

a-=l  +(log,a)  a;+  (log,a)2|j  +  -.  (1) 

Substituting  in  this  1  +  y  for  a,  we  have 

(l+2,)-  =  l  +  a;logXl+y)  +  5(log,(l+y))2+....        (2) 

If  y  is  numerically  less  than  unity,  we  can  expand  by  the 
binomial  theorem  and  get 

-,    ,         ,  x(x—X)    2  ,  x(x  —  X)(x  —  ^^    q  , 

i  +  ^^  +  -^-2| — -y  -^— ^7 -y^-^'" 

=  \^x  log/l  +  y)  +  ^  (log,(l  +  y))2  +  ....      (3) 

Equating  the  coefficients  of  x  on  both  sides  of  (3),  we 
have  (212,  17) 

log,(l  +  ^)=y-f  +  f-^^V-.  (4) 

This  is  called  the  logarithmic  series. 
Changing  y  into  —  ^,  we  have 

log,(l-^)= -^-|--|--^ .  (5) 


242  COLLEGE   ALGEBRA 

The  logarithmic  series  may  be  used  to  find  the  logarithm 
of  any  number,  but  since  the  series  converges  so  slowly,  it  is 
more  expedient  to  use  the  following: 

Subtracting  the  corresponding  members  of  (4)  and  (5), 
we  have 

ioge(i  +  y')  -  ioge(i  -y^=\y~\'^\  — ) 

\     -^      2        3 
log,J-±^=2(y  +  |  +  ^+...).  (6) 

^       T       I      7/  /yyi  fvyj    'Y) 

Substitute  in  this ^  =  — ,  that  is,  y  = ,  and  it  becomes 

1  —  y      n  m-\-n 

^      rn^^jm-n\^m-n\^^ljm-n\^^  ...Y       (7) 
n         \m  4-  n      3  \m  +  7iJ       5  \m  +  nj  •' 

If  m  =  n^  logg  1  =  0. 

If.  =  2a„d«  =  l,log.2  =  2g  +  lg)Vlg)%. 

If  m  =  n  +  1,  (7)  becomes 


log.  '±^  =  2{,,^^  +  i(,r^^]+iJ,r^^]  + 


n         "\2n+l      iy2,n  +  lj      5V2»  +  ] 


and  this  is  equivalent  to 


1^     1     ^V-l-         (8) 


5\2n-\-  1 


LOGARITHMS  243 

From  (8)  the  logarithms  of  all  numbers  to  the  base  e  may 
be  obtained.  It  has  been  shown  in  271  how  to  change  from 
one  base  to  another. 

To  obtain  the  logarithms  of  numbers  to  the  base  10  we 
have  to  multiply  the  logarithms  of  the  numbers  to  the  base  e 

by ,  which  may  be  found  from  (8)  to  be  0.434294+ . 

log,  10 

Examples.     Compute  log^  2,  log, 3,  loge4,  logjQ2,  logio3» 


CHAPTER  XVII 

DETERMINANTS 

286.  If  in  algebra  certain  forms  are  of  frequent  occurrence, 
it  is  convenient  to  have  a  suitable  notation  to  express  tliem; 
thus  the  expressions,  a^^  —  aj)-^^  ^i^2^3  ~l~  ^j^2pi  +  ^3^i^2  ~  ^3^2^! 
—  a^^c^  —  a^^c^^  are  instances  of  such  forms  which  often 
occur.    The  first  may  arise  as  the  result  of  eliminating  x  and 

y  from  the  equations 

a^x  +  a<2^y  =  0, 

h^x  4-  h^^y  =  0. 

The  second  from  eliminating  x^  ?/,  z  from  the  equations 

a^x  4-  ci^y  +  ^3^  ~  ^' 
h^x  +  h^y  +  h^z  =  0, 
c^x  +  c^y  4-  c^z  =  0. 

In  general,  if  we  eliminate  the  n  variables  from  a  system 

of  n  linear  homogeneous  equations,  we  get  a  result  of  the 

form  _         ,  7        /x 

z  ±  a-^o^c^  ■•'I,,  =  0. 

These  forms  are  called  determinants^  and  since  they  are 
functions  of  the  coefficients  only  and  may  be  considered 
without  reference  to  their  origin,  the  preceding  being  but 
one  way  in  which  they  arise,  the  definition  of  a  determinant 
should  obviously  contain  no  reference  either  to  its  origin  or 
to  the  variables. 

244 


DETERMTXANTS 


245 


287.  Definition.  If  tve  have  ^t^  quantities  arranged  in  a 
square  of  n  roivs  and  n  columns^  then  the  sum^  with  j^roper  signs 
of  all  the  terms  that  can  he  formed  by  taking  the  product  of  n 
quantities  one  and  ojily  one  from  each  row  and  one  and  only 
one  from  each  column^  is  called  the  determinant  of  those  quanti- 
ties and  is  said  to  be  of  the  nth  order. 

The  sign  factor  for  any  term  of  the  determinant  as  defined 
above  is  determined  by  writing  in  succession  the  numbers  of 
the  rows  from  which  the  quantities  composing  it  have  come; 
and  in  a  separate  series  the  numbers  of  the  columns  and  tak- 
ing +  or  —  according  as  the  total  number  of  inversions*  of 
order  in  the  two  series  is  even  or  odd.  Since  the  factors  of 
any  term  may  be  written  in  any  order  whatever,  we  may 
obviously  write  them  so  that  the  numbers  in  one  of  these  two 
series  are  in  the  natural  order  (order  of  magnitude)  and 
therefore  in  determining  the  sign  factor  of  a  term  we  need 
only  take  account  of  the  inversions  in  the  other  series.f 

288.  Notations  and  Definitions.  The  ordinary  notation 
for  a  determinant  is 


1      ^*> 

^1   h 


for  «j^2  ~  ^2^1  ' 


a^  a^ 

^3 

h    K 

^3 

^1     ^2 

^3 

iova^K^c^+aJ)^c^  + 


«1   «2    ■ 

^1    ^2    •• 

'l     h     ■ 

■■  i„ 

for  S  ±  a^^c^'-'l^. 


*  Whenever  a  greater  integer  precedes  a  less  there  is  said  to  be  an  inver- 
sion of  order. 

t  Show  that  the  same  sign  factor  results  whatever  be  the  order  of  the 
series,  pairs  being  kept  together. 


246  COLLEGE   ALGEBRA 

The  quantities  a^  a^,  etc.,  are  called  the  constituents  or  ele- 
ments^ and  the  products  a-J)^,  (t-^b^c^,  etc.,  are  called  the  terms 
of  the  determinant. 

In  the  square  array  representing  a  determinant  the  diag- 
onal from  the  left-hand  top  corner  to  the  right-hand  bottom 
corner  is  called  the  principal  diagonal;  and  the  diagonal 
from  the  left-hand  bottom  corner  to  the  right-hand  top 
corner  is  called  the  secondary  diagonal.  The  term  formed 
by  the  product  of  all  the  constituents  along  the  principal 
diagonal  is  called  the  principal  or  leading  term. 

Another  convenient  notation  for  determinants  is  to  write 
each  constituent  with  a  double  sufiix,  the  first  indicating  the 
row  and  the  second  the  column  to  which  the  constituent 
belongs.  Thus  the  determinant  of  the  ni\\  order  in  this 
notation  is  written 


^21        ^^22        ' '  *        ^2't 


,  or  simply  (a^^  a^^  •••  a„„). 


Elements  are  said  to  be  co7ijugate  to  each  other  when  the 
place  that  either  occupies  in  the  row  is  the  same  as  the 
other  occupies  in  the  column.  Thus  ajj.  and  aj^^  are  conju- 
gate elements.  Elements  along  the  principal  diagonal  are 
self- conjugate. 

289.  The  principal  term  is  a-^-^  a^^  a^^  a^^  '"(^nn^  and  we 
can  get  all  the  other  terms  from  this  by  interchanging  the 
second  (or  the  first)  series  of  suffixes  in  all  possible  ways, 
for  this  gives  all  possible  ways  of  taking  one  and  only  one 
element  from  each  row  and  column.  If  we  take  any  term 
a^i^  ^2,-  %  •••  a„i^  where  the  fs  form  a  permutation  of  the 
numbers  1,  2,  3,  4,   •••  n^  and  interchange  two  adjacent  fs, 


DETERMINANTS 


247 


we  obtain  another  term  of  the  determinant  which  has  a  sign 
opposite  to  that  of  the  given  term,  for  we  have  either  one 
more  or  one  less  inversion  among  the  is.  If  any  two  fs 
having  r  suffixes  between  them  are  interchanged,  the  result- 
ing term  will  have  a  sign  opposite  to  the  given  term,  for  the 
interchange  can  evidently  be  brought  about  by  2  r  -h  1  suc- 
cessive interchanges. 

EXAMPLES 


1.    Determine  the  sign  of  ^j^  ^22  ^33 


a^^  considered  as  a 


term  of 


a 


a 


11 

31 

^41 


12 
^22 
'32 
i^42 


a 


a 


a 


13 

33 
43 


a 


The  constitutents  are  already  written  down  in  the  order 
of  their  rows.  The  series  of  numbers  denoting  the  columns 
from  which  the  constituents  have  come  is  1,  2,  3,  4,  and 
contains  no  inversions  of  order.  The  sign  of  the  term  is 
therefore  +. 

2.  For  the  same  determinant  find  the  sign  of  a^^  a^i  a^2  ^w 
To  determine  the  sign  in  this  case  we  have  to  determine 
the  number  of  inversions  of  order  in  the  series  3,  1,  2,  4. 
As  there  are  two  inversions  of  order,  viz.  3  before  1  and  3 
before  2,  the  sign  is  + . 

3.  For  the  same  determinant  determine  the  signs  of  tlie 


following  terms  :  a^^  a^i  a^^  ^43  ;  <^i2  ^21  ^ 


34  ^43  '  ^ 


13  ^^24  '^31  ^ 


42* 


4.    For  the  determinant 


^1 

«2 

h 

h 

^1 

^2 

d. 

cL 

a. 


do    d. 


248  COLLEGE   ALGEBRA 

determine  the  sign  of  the  term  a^  h^  c^  dy  As  the  elements 
are  arranged  in  their  natural  row  order  we  have  the  series, 
3,  2,  4,  1,  from  which  to  determine  'the  sign.  There  are 
three  inversions  and  the  sign  is  therefore  — . 

5.  For  the  same  determinant  as  in  the  last  example  find 
the  signs  of  the  terms  ^2  ^4  ^3  ^1 5  ^3  ^1  ^2  ^4 '  ^4  ^1  ^3  ^2' 

6.  Find  all  the  terms  of  the  determinant 


^1 

^2 

^3 

h 

h. 

^3 

^1 

^2 

^3 

290.  Theorem.  The  iiumhei'  of  terms  of  a  determinant  of 
the  nth  order  is  n  !.  This  follows  at  once  from  the  fact  that 
each  term  contains  one  constituent  from  each  row  and  one 
constituent  from  each  column,  and  therefore  there  are 
as  many  terms  as  there  are  arrangements  of  n  things  all 
together. 

291.  Theorem.  In  any  determinant  there  are  as  many  posi- 
tive as  there  are  yiegative  terms;  for  if  we  interchange  two 
suffixes  of  tlie  second  series  in  a  positive  term,  we  get  a 
negative  term,  and  therefore  there  are  as  many  or  more 
negative  terms  than  there  are  positive  terms,  and  the  inter- 
change of  any  two  suffixes  in  a  negative  term  will  give  a 
positive  term,  and  there  are  as  many  or  more  positive  terms 
than  there  are  negative  terms,  and  therefore  there  must  be 
the  same  number  of  each. 

292.  Theorem.  If  every  element  of  a  row  or  column  of  a 
determinant  is  zero,  the  determinant  is  zero.  This  follows 
from  the  fact  that  every  term  of  the  determinant  contains 
one  element  from  this  row  or  column. 


DETERMINANTS  249 

293.  Theorem.  Two  determinants  tvhich  differ  only  in  hav- 
ing the  rows  of  one  the  same  as  the  cor  responding  columns  of  the 
other  are  equal.  Every  term  of  the  one  determinant  contains 
an  element  from  each  row  and  each  column  of  that  deter- 
minant, and  therefore  it  contains  an  element  from  each 
column  and  row  of  the  other  determinant,  and  therefore  is 
a  term  of  the  other  determinant.  That  the  sign  factor  of 
these  two  terms  is  the  same  is  evident,  since  it  is  determined 
from  the  same  series  of  numbers  in  each  case. 

From  this  it  follows  that  in  any  proposition  involving  the 
terms  "  row "  or  "  column  "  we  may  get  another  which  is 
equally  true  by  substituting  the  terms  "  column  "  or  "  row  " 
respectively. 


1.    Show  that 


EXAMPLES 


«1 

«2 

H 

^1 

h 

h 

= 

H 

H 

^3 

a^     ^j     c^ 

^2         2       ^2 

a^     O3     c^ 

2.  Write  the  negative  terms  of  the  determinant  in  the 
preceding  example. 

294.  Theorem.  If  tivo  columns  of  a  determinant  he  inter- 
changed^ the  resulting  determinant  differs  only  in  sign  from 
the  given  determinant;  for  this  amounts  to  an  interchange  of 
two  of  the  second  set  of  suffixes  in  each  term  and  therefore 
to  a  change  of  sign  of  that  term  ;  consequently  the  sign  of 
the  whole  determinant  is  changed.  ' 

295.  Theorem.  Jf  two  roivs  of  a  determinant  he  identical., 
the  determinant  is  equal  to  zero. 

Let  the  determinant  be  A,  then  interchanging  these  two 
rows,  we  have  a  determinant  which  by  the  preceding  article 


250 


COLLEGE   ALGEBRA 


is  equal  to  —  A,  but  the  resulting  determinant  is  exactly  the 
same  as  A  on  account  of  the  identity  of  the  two  rows,  there- 
fore 

A, 


or 


A  = 
A  =  0. 


^11 

«12   • 

••    «1«, 

«21 

^22  ' 

••    «2« 

^nl 

^«2   •• 

^nn 

296.  Since  by  definition  every  term  of  a  determinant 
contains  one  and  only  one  constituent  from  each  row  and 
column,  it  follows  that  a  determinant  is  a  linear  homogeneous 
function  of  the  constituents  of  any  row  or  column.     Thus 


~  ^11^11  "I"  ^12^12  +  *^13^13  +    '**    +  ^1«^1»' 


where  the  ^'s  contain  no  constituent  from  the  first  row. 

297.  Theorem.  If  all  the  constituents  in  any  roiv  he  multi- 
plied by  the  same  number,  the  resulting  determinant  is  equal  to 
the  product  of  the  original  determinant  and  this  number. 

For  if  we  expand  the  determinant  as  a  linear  function  of 
the  elements  in  this  row,  the  given  number  will  appear  as  a 
factor  of  every  term  and  therefore  of  the  determinant. 

298.  Theorem.  If  the  constituents  of  any  row  differ  from 
those  of  any  other  row  by  the  same  constant  factor^  the  deter- 
minant vanishes. 

For  taking  out  the  common  factor,  there  results  a  deter- 
minant with  two  indentical  rows,  and  this  is  equal  to  zero. 

Example.     In  the  determinant 

2  5  6 
3-3  9 
14     3 


DETERMINANTS 


251 


the  elements  of  the  hist  column  are  three  times  the  corre- 
sponding elements  of  the  first  column  and  therefore 

5 


=  3 


2  5  2 
3-3  8 
1        4     1 


=  0. 


299.  Theorem.  If  each  of  the  constituents  of  a  row  of  a 
determinant  consists  of  tivo  terms^  the  determinant  ynay  he  ex- 
pressed as  the  sum  of  two  determijiants. 

For  if  the  determinant  be 


A  = 


an  +  ^\i  «i2  +  ^^i2  •••  'hn  +  K 


«21 

a,^i 


22 


^0 


^n2 


a, 


expanding  in  terms  of  the  elements  of  the  first  row  we  have, 
A  =  («ii  +  ^i)^ii  + (^^12  +  ^12)^12+  •••  +(^i«  +  ^i«Mi« 

=  ^1^11  +  ^12^2+  •••  +  ^l«^l«  + ^11^11 +  ^12^2+  •••  +hnAn 


^11 

<^12  • 

'  (^m 

^11 

^^12   • 

••   Ki 

«2i 

«22   • 

•   (l2n 

+ 

«21 

^22    ■ 

■'   (hn 

<^nl 

<^^n2   ' 

'   ^nn 

f/„l 

««2   • 

■   ^nn 

300.  Theorem.  If  each  of  the  constituents  of  a  roiv  is  equal 
to  the  sum  of  r  terms,  it  is  obvious  that  the  determinant  is 
equal  to  the  sum  of  r  determinants.  This  may  be  farther 
generalized  by  having  polynomials  for  the  constituents  of 
other  rows. 

301.  Theorem.  The  result  of  299  may  be  viewed  as  a 
theorem  for  the  addition  of  two  determinants,  giving  the 


252 


COLLEGE   ALGEBRA 


theorem  :  If  two  determinants  are  alike  except  as  regards  the 
elements  in  the  rth  row  of  each^  their  sum  is  equal  to  a  de- 
terminant which  is  like  each  of  the  others  except  that  any 
constituent  of  the  rth  row  is  the  sum  of  the  corresponding 
constituents  of  the  two  given  determinants.  This  theorem  also 
may  be  generalized. 

302.  Theorem.  If  the  constituents  of  any  row  he  increased 
(^algebraically^  hy  equimultiples  of  the  corresponding  con- 
stituents of  any  other  row^  the  determinant  is  unaltered. 

For  the  resulting  determinant  is  equal  to  the  sum  of  two 
determinants,  one  of  which  is  the  original  determinant  and 
the  other  is  this  multiple  times  another  determinant  having 
two  rows  identical,  and  therefore  vanishes. 

The  principle  of  this  article  is  useful  for  simplifying  and 
evaluating  determinants  whose  constituents  are  numerical. 
Thus  if  in  the  determinant 

2        13 

-3    -2    -4 


we  add  the  elements  of  the  second  row  to  those  of  the  third, 

we  have 

2        13 

-3    -2  -4 

0-1       1 


and  in  this  if  we  add  two  times  the  elements  in  the  first  row 
to  those  of  the  second,  we  get 


2  1 
1  0 
0   -1 


DETERMINANTS 


253 


and  finally  if  in  this  last  form  we  add  the  elements  of  the 
second  to  those  of  the  third  column,  we  get 


2 

1     4 

1 

0     2 

0 

-1     0 

which  when  the  common  factor  2  is  removed  from  the  third 
column  is  seen  to  vanish,  having  two  identical  columns. 


303.  EXAMPLES 

1.    Show  that  a -\- h  -{-  c  is  'd  factor  of 

a     h     h  -\-  c 


c  -\-  a 
a  -\-b 


9 

a  —  mg     d    g 

a     d    g 

h 

= 

h  —  mh     e     h 

= 

h      e     h 

k 

c  —  mk    f     k 

c      f     k 

2.  Show  that 

a  +  md  d 
h  -i-rne  e 
c  +  mf     f 

3.  Write  in  determinant  form  the  following  expressions 

(1)  ahc  +  2  hgf  -  g%  -  ¥c  -f^a, 

(2)  3  xyz  —  Q^  —  y^  —  z^.     , 


4.    Show  that 


5.    Prove 


ho 

a 

ca 

h 

ah 

e 

a^ 

n1 


2  1-9 

8-5         4 

12         7-3 


a'' 


119 

=  2 

4     5     4 

6     7     3 

304.    Minors.    If  we  delete  a  rov/  and  a  column  of  a  deter- 
minant, the  determinant  of  the  remaining  (ii  —  1)^  elements 


254 


COLLEGE   ALGEBRA 


is  called  a  minor  of  the  (w  —  l)th  order  of  the  original  deter- 
minant, or  a  first  minor. 
From  296  it  is  seen  that 


A  = 


a, 


11 

21 


12 
''^22 


'in 


''2H 


a 


m 


a 


«2 


a. 


=  ^.1^11  +  ^10^10+    ••.    +«i„4 


"12^-^12 


In^^lni 


and  that  the  expressions  A-^^  A-^^^  •••,  A-^^  contain  no  elements 
from  the  first  row.  What  these  expressions  are  may  be  seen 
on  referring  to  the  definition.  For  the  coefficient  of  6Kjj  by 
definition  must  contain  one  and  only  one  constituent  from 
each  of  the  other  rows  and  columns  except  the  first,  and  there- 
fore can  be  nothing  more  than  the  minor  formed  by  deleting 
the  first  row  and  column.  In  the  same  way  after  passing 
the  second  column  over  the  first,  which  changes  the  sign  of 
the  determinant,  we  find  that  A-^^^  the  coefficient  of  a^^^  is  the 
negative  of  the  minor  obtained  by  deleting  the  first  row  and 
second  column  ;  similarly  ^j 3,  J.J5,  •••  are  the  minors  formed 
•by  deleting  the  first  row  and  third  column,  first  row  and  fifth 
column,  etc.,  and  ^j^,  ^jg,  •••  are  the  negatives  of  the  minors 
obtained  by  deleting  the  first  row  and  fourth  column,  the 
first  row  and  sixth  column,  etc. 

If  we  denote  by  A^j,  the  minor  formed  from  A  by  deleting 
the  ith.  row  and  ^th  column,  and  expand  A  in  terms  of  the 
elements  of  the  ith  row,  we  have 

A=(-iy-V.Ai-«.2^2+  •••  +(-iy^-KA,j. 

Using  this  principle  to  expand  the  determinant 


A  = 


a 


a. 


a 


12 


ar 


22 
31       ^32 

a,,     a 


a 


"41 


42 


a 


a 


■13 
23 

^33 
43 


a 


a 


24 
^34 
44 


DETERMINANTS 


255 


we  have       A  =  —  «52i^2i  +  ^^22^22  "~  ^23^23  +  ^24^24 

=  ^31^3^  —  ^32^32  +  ^'^33^33  —  <^^34A34 

=  -  ^14^14  +  ^24^24  -  ^^34^34  +  ^^44^44' 

Applying  this  to  a  numerical  example,  we  have  for  the 
determinant 


1 

2 

3 

2 

5 

4 

3 

7 

8 

5 

2 

6 

1 

6 
5 
3 

5    4    6 

2    4    6 

2    5    6 

2    5    4 

= 

7    8    5 

-2 

3    8    5+3375 

— 

3    7    8 

2    6    3 

5    6    3 

5    2    3 

5    2    6 

5    4    6 

5    4    2 

2    4    6 

2    5    61 

7    8    5 

— 

7    8    3 

-2 

3    8    5+3375 

2    6    3 

2    6    5 

5    6    3 

5    2    3 

5    4 

4 

2 

7 

= 

7    8 

2 

+ 

3 

5 

2    6 

-2 

5 

-6 

— 

-20. 

(301,  302) 


305.    The  expression 

'^31^21    '    ^32     22    ■"  '^33'^23    ■"  ^34^24' 

which  is  obtained  from  an  expansion  of  the  determinant  A  of 
304,  by  putting  the  elements  of  the  third  line 

in  the  place  of 


'31'  '■*32'  "'33''  ^^34' 
^2V  ^22'  ^23'  ^24' 


respectively,  obviously  vanishes,  since  it  is  equivalent  to  the 
determinant 


'11 

^31 


^2       ^13 


a 


^32 


'33 


14 
^34 


^31       ^32       ^*^33       ^34 


a 


41 


'i2 


hs       ^44 


in  which  the  second  and  third  rows  are  identical. 


256 


COLLEGE   ALGEBRA 


Thus  in  general  if  we  multiply  the  elements  in  any  row  of  a 
determinant  hy  the  corresponding  cof actors  of  the  elements  of 
any  other  row^  the  sum  of  the  products  thus  formed  is  equal  to 


zero. 


306.    If  we  multiply  the  three  equations 
a^-^x  +  ay^y  +  ^132;  =  ^14, 

a^^x  +  ^32^  +  a^^z  =  ^34, 

by  A^-^,  ^21'  ^3P  resjDectively,  and  add  them,  we  have 
(^11^11  +  ^21^21  +  ^31^31)^^  +  (^12^11  +  ^22^21  +  ^32^31)^ 

+  («13^11  +  «23^21  +  ^33^31)^  =  «14^11  +  ^24^21  +  «34^31- 

If  we  denote  by  A  the  determinant 


11  1^ 


13 


a. 


21 


a 


a 


31 


23 


^32     ^33 


whose  elements  are  the  coefficients  of  x,  y,  z,  in  the  three 
given  equations,  it  will  be  seen  that  the  coefhcient  of  x  in 
the  fourth  equation  is  A,  that  the  coefficients  of  y  and  z 
vanish,  and  the  right-hand  member  of  the  equation  is  the 
determinant  formed  by  writing  a^^,  a^^,  a^^,  in  place  of  the 
elements  «j^,  a^i^  a^^  in  A. 
We  have  therefore 


x  = 


«14 

^12 

«13 

^24 

^22 

^23 

%4 

'^32 

^33 

DETERMINANTS 


257 


Similarly 


y  = 


z  = 


%1 


'14 


24 


a 


34 


^13 


•23 


a 


33 


A 

^^11 

«12 

^^14 

«21 

«22 

^^24 

«31 

^32 

%4 

SO  that  in  general  three  linear  non-homogeneous  equations 
in  three  unknowns  have  one  and  but  one  set  of  values  of 
X,  y^  z  which  satisfies  them. 


EXAMPLES 


1.    Solve  the  equations 


X  —  \y  =  ^. 


We  have 


x=  - 


5 

3 

(3 

-4 

2 

3 

1 

-4 

—  3  8. 
"""  11' 


and 


y  = 


2 

5 

1 

G 

2 

3 

1 

-4 

7_ 

~"       11' 


2.    Solve  the  equations 

2a;-3^  +  ^  =  7, 
x-\y-\-±z  =  ^, 
3a;-f-y— 3s  =  —  4. 


258  COLLEGE   ALGEBRA 

307.  If  in  the  equations  of  the  preceding  article  a-^^  =  a^^  = 
a^^  —  0,  and  therefore  the  equations  are  homogeneous,  it  fol- 
lows that  if  we  are  to  have  values  of  x^  ?/,  and  z  other  than 
zero,  which  satisfy  the  equations,  A  must  be  equal  to  zero. 
That  is,  to  have  a  set  of  three  linear  homogeneous  equatio7is  in 
three  unknown  quantities^  satisfied  hy  values  other  than  zero^  the 
determinant  of  the  system  must  vanish. 

This  is  evidently  just  as  true  for  n  linear  homogeneous 
equations  in  n  unknowns  as  for  three. 

Example.    Show  that  the  three  equations 
ax+{h  +  c)y-\-z^^, 
hx-\-(c-\-  a^y  +  3  =  0, 
ex  +  (a  +  J)?/  +  3  =  0, 
are  satisfied  by  values  of  x^  ?/,  z  other  than  zero. 

308.  If  we  divide  the  three  linear  homogeneous  equations 
of  the  preceding  article  hj  z  (z4^  0),  and  put 


X 

-  =  u, 

z 

and 

z 

we  get 

a^^u  +  a^^v  +  fl'i3  =  0, 

a^^u  +  a^^v  4-  a^^  =  0, 

a^^u  +  a^^^v  +  «33  =  0, 

that  is,  three  equations  with  two  unknown  quantities,  u  and  v. 
But  two  independent  non-homogeneous  linear  equations  de- 
termine a  unique  set  of  values  of  u  and  v^  and  there  is  no 
third  equation  which  is  satisfied  by  these  same  values,  unless 


DETERMINANTS 


259 


it  is  dependent  upon  the  other  two  equations.  Solving  the 
first  two  for  u  and  v  and  substituting  these  values  in  the 
third,  we  get  A  =  0,  wliich  coincides  with  what  we  have  just 
seen,  viz.  tliat  A  must  vanish  in  order  to  have  the  three  equa- 
tions consistent.     In  fact  A  =  0  is  satisfied  if  we  take 

ttq9  —   ^19  ~T~   '^^^22' 


a    =  a-,o-\-  \a 


^^33 


13 


^23' 


that  is,  the  third  equation  is 


^11 


u  +  a^^v  +  rt^3  H-  \(^a.2{ii  +  ^22^  +  ^23)  ==  ^' 


or  it  is  equivalent  to  the  first  plus  X  times  the  second  ex- 
pression equated  to  zero. 


309.    Product  of  Two  Determinants.     Let 


A"  = 


a 


11 


'21 


12 


22 


'13 


•^23 


^31   ^^32   ^33 


,  A'  = 


'11 


^12   ^ 

^21   ^^22   ^ 
h 


13 


23 


'31 


'32 


33 


^11^11  +  ^12^2  +  ^13^3  ^21^^11  +  ^^22^12  +  ^^23^13 
^11  21  ~^  ^^12^22  ~^  ^13  23  ^21  21  ~^  ^22  22  "•"  ^*23  23 
^11^31  +  ^12^32  +  ^13^33   ^'^21^31  +  ^22^32  +  ^23^33 


^3Al  +  ^^32^12 +  ^^33^3 
^31  21  •"  ^32  22  "■  ^33^23 
%1^31  ■^"  ^32^32  +  ^33^33 

The  determinant  A^'  may  be  partitioned  into  twenty-seven 
determinants  of  the  third  order  with  monomial  elements. 
Twenty-one  of  these  determinants  vanish  identically,  having, 


260 


COLLEGE   ALGEBRA 


after   common  factors   are  removed,  two  or  more  identical 
columns.     Thus  one  of  them  is  the  determinant 


^11^11       %l*^ll       ^^32*^12 

11     91  ^91     91  ('oc}0()(f 


^11^31       ^21^31 


=  a 


11 


a. 


21 


'32 


^11       ^11 
^21       ^21       ^^ 


'12 


22 


'31 


'31 


32 


=  0. 


The  determinants  which  do  not  vanish  identically  are  the 
following : 


^11^1 

^22    12 

%3'^13 

^11^11 

^23-13 

^32^12 

'^11^21 

^33*^23 

1 

^11^21 

^23^23 

^^32    22 

«11^31 

^22    32 

<^33''33 

^11^^31 

^23^33 

''*^32    32 

%2    12 

^21^11 

^33*^13 

'^12^12 

^23^13 

hAi 

^12    22 

^21^21 

^33^23 

1 

^12   22 

^23   23 

HxK 

^^12   32 

^21^31 

^33^33 

'^12*^32 

^23''33 

'hAi 

«13^^13 

%1^11 

^*32'^r2 

^13-13 

^'I'Pvi 

^31^11 

'^13^23 

^21^21 

^^32    22 

^ 

^^13^23 

^22    22 

a^iO^i 

^13^33 

^21^31 

^32^32 

^13^^33 

^22    32 

^31^31 

Talking  out  the  common  factors,  we  have 

A      =  A  '^ii^22'^33        ^  ^11*^23^32  ~  ^  ^12*^21^33  "•    ^  '^12^23^^31 

+  A'a^^a.^^(u^  -  A^)'igrt22%  =  AA'. 

It  will  be  observed  that  to  form  a  determinant  which  does 
not  vanish  identically,  the  first  column  may  be  chosen  in 
three  ways ;  and  when  tliat  is  chosen,  the  second  may  be 
chosen  in  two  ways,  and  the  third  in  one  way,  making  in  all 
six  determinants  which  do  not  vanish  identically. 

The  student  may  state  the  rule  for  the  multiplication  of 
two  determinants. 


DETERMINANTS 


261 


Example.    Write  as  a  determinant  the  square  of 


X 


•/-r 


1       -^2       -^3 

yi  Vi  Vz 

^1        ^2        ^3 


310.  EXAMPLES 

1.    Evaluate  the  determinant 


3 

4 

6 

7 

5 

4 

9 

8 

1 

2 

7 

3 

0 

5 

3 

0 

2.    Expand 


a     h 

0 

h     h 

f 

g  f 

c 

3. 

Expand 

0 

a     h     e 

a 

ode 

h 

d     0    f 

c 

e    f    0 

Expand  in  terms  of  the 
elements  of  the  4th  row. 


4.    Evaluate 


4 

9 

2 

3 

5 

7 

8 

1 

6 

The  numbers  in  this  determinant  are  arranged  in  what 
is  known  as  a  magic  square.  The  sum  of  the  elements  in 
any  line  is  15. 

5.    Solve  the  equations 

x  +  2y  +  z  =  A, 

5x— 1/  —  oz  =  l^ 

4a:  +  ^  —  2  =  3. 


CHAPTER   XVIII 
THEORY  OF  EQUATIONS 

311.  We  have  now  to  consider  the  general  properties  of 
the  polynomial  with  real  coefficients, 

f(x)  =  a^x""  +  ^ia^"~i  +  ^2^""2  +  •••  4-  a^''-^  +  •••  +  «,„  (1) 

and  those  values  of  x  which  make  it  vanish. 

312.  A  question  which  naturally  arises  concerning  the 
polynomial  is,  which  are  the  significant  terms  in  the  case 
of  large  or  small  values  of  x. 

In  order  that 

a^x^  >  a^x^~'^  +  a^x^~^  +  •  •  •  +  a^,  we  must  have 


a-,x^  ^  -{-■'•  -\-  a. 


Dividing  numerator  and  denominator  of  the  left  member  of 
the  inequality  by  x''~'^,  we  must  have 


^0^1  __._  ->  i 


Li,  f>  t*^. 

a. +^+  •••  + 
^       x  x 


,11— \ 


(3) 


313.  As  X  becomes  indefinitely  great,  the  left  member  of 
the  inequality  becomes  indefinitely  great,  and  therefore  for 
some  value  of  a;,  as  x  increases,  the  fraction  must  be  greater 
than  one  ;  that  is,  as  x  increases  there  is  a  value  of  x  for 
which  and  for  greater  values, 

a^^x^  >  a^x^~'^  +  a^x^~'^  + [-  a^,  (4) 

262 


THEORY   OF   EQUATIONS  263 

314.  By  what  has  just  been  proven, 

for  sufficiently  great  values  of  y,  that  is 

a^     «^  «5  (6) 

y       y  y 

or  II  -  =  a;, 

y 

for  sufficiently  small  values  of  x.  Therefore  the  independent 
term  is  greater  than  all  the  others  if  x  is  taken  sufficiently  small. 

315.  The  series  2_i_      _i_       n 

may  he  made  as  small  as  we  please  hy  making  x  sufficiently 
small,  for  it  may  be  made  smaller  than  any  assigned  number, 
however  small,  which  may  be  taken  as  a„. 

316.  In  the  same  series  the  sign  of  the  series  may  he  made 

to  depend  upon  that  of  the  term  a^-^x^  for  the  series  may  be 

written  .  „_j. 

X\a^_-^  -f-  aj^_^ -f-  •••  +  a^      j^ 

whence  the  statement  is  evident. 

317.  Development  of  a  Function.     If  in  /(a;)  we  put  x-{-h 
for  x^  we  have 

f(x  +  A)  =  a^^x  +  hy  +  a^Qx  +  A)"-i  +  •••  +  a„_i(x  +  A)  +  a„ 
=  a^x"  +  a^x''-'^  H \-a„+\  ^la^x''-'^  +  {n  —  l^a^x""'^ 

+  •••  +  2  a,,_^x  +  a^_-^lh  +  ^  ln(n  -  l^a^x^-^-  +  0^  -  l)^^  -  2) 

«i:r"-3  4-  ...  +  3  .  2  .  a,,_^x  +  2  a,Jh'^+       _^  n^i  -  1)  ..-l     ,^. 

n ! 


264  COLLEGE   ALGEBRA 

or  denoting  the  coefficient  of  Jf  by      - — ^-^, 

fCx  +  h)  =fix)  +  hfix)  +  ^J"ix-)  +  ...  +  ^n-Kx-)  . 

318.  The  different  expressions/'(a:),/''(2;),  etc.,  are  called 
the^rs^,  second^  etc.,  derived  functions  of  x.  It  will  be  seen 
that /'(a;)  is  formed  from  /(a;)  by  multiplying  each  term  in 
f(x)  by  the  exponent  of  x  in  that  term,  diminishing  the 
exponent  (of  x^  by  one,  and  taking  the  sum  of  all  such 
terms.  Again /"(a;)  is  the  first  derived  function  oif'(x')^ 
and  in  general /^^^ (a;)  is  the  first  derived  function  oif^'~'^\x^. 

EXAMPLES 

1.  Given  f(x)  =^  x^ -1  x^  +  x-1,  find  fQx  +  2). 
We  have  for  the  derived  functions 

f'Qx)  =  S)x^-4:x  +  l, 
f"{x-)  =  lSx~4, 
f"'(x)  =  lS. 

Using  X  for  7i,  and  2  for  x  in  the  foregoing  formula,  we  have 

/(2)  =  17, 

/'(2)  =  29, 

/"(2)  =  32, 

/"'(2)=18. 

Therefore      f(x+2}  =  11  +  29  x  +  16  x^  +  S  x?. 

2.  Given /(a:)  the  same  as  in  the  last  example,  find 

f(x  +  l),f<:x-2-),fCx-l). 


THEORY  OF  EQUATIONS 


265 


319.    From  the  above  development  for/(a:+  li)  we  get 
fix  +  70  -/(x)  =  hf'ix)  +  ^/"  (:,)  +  ...  +  ^  /W(^). 

By  316,  /(:2:^+  A)  —f(x)  may  be  made  less  than  any  assigned 
number  by  taking  h  sufficiently  small.     This  means  that  if  x 


Fig.  27. 

increases  by  indefinitely  small  increments,  from  x^  to  x^^  /(^) 
changes  by  indefinitely  small  increments  from  f(x^  to  f(x^)^ 
or  f(x)  is  said  to  vary  continuously 
between  x^  and  x^.  As  a  conse- 
quence of  this,  it  is  readily  seen 
that  if  f(x^  and  f{x^  have  oppo- 
site signs,  there  must  be  some  value 
of  X  lying  between  x^  and  x^^  such 
that/ (a:)  vanishes  for  this  value. 
The  graphical  significance  of  con- 
tinuity is  that  the  curve  y—f{x^ 
is  uninterrupted  between 
any  two  of  its  points  as  in 
figure  27,  and  cannot  be 
interrupted  as  in  figure  28.  Fig,  28. 


266  COLLEGE   ALGEBRA 

320.  Theorem.  Every  function  of  odd  degree  vanishes  for 
a  real  value  of  x  which  has  a  sign  opposite  to  that  of  a^,  for  by 
substituting  —  oo,  0,  and  +qo  for  a;,  the  function  is  negative, 
has  the  sign  of  a„,  and  is  positive  respectively,  and  therefore, 
by  319,  vanishes  for  a  value  of  x  opposite  in  sign  to  that  of 
a„,  for  the  function  will  change  sign  between  —  go  and  0,  or 
between  0  and  +go  according  as  a^  is  positive  or  negative. 

321.  Theorem.  Every  function  of  even  degree  which  has  a^ 
negative  vanishes  for  one  positive  and  one  negative  value,  for 
this  function  changes  sign  between  — oo  and  0,  and  between 
0  and  +  Qo .  This  is  seen  graphically  in  that  to  change  sign 
the  graph  representing  the  function  must  cross  the  axis  of  x, 
hence  there  must  be  a  root  for  the  value  of  x  at  which  it 
crosses. 

EXAMPLES 

1.  Show  that  x^  —  2x'^-{-5=0  has  a  negative  root. 

2.  Show  that  x^— 2x^  +  4: x^— Sx  — 2  =  0  has  at  least  a 
positive  root  and  a  negative  root. 

322.  In  81  we  have  shown  that  if  a^^  0, 

f(x)  =  a^^x  -  ccj) (x-a^}  -'•  (x-  a„), 

where  a^,  «2i  •••  «„  are  the  roots  of /(a;)  =  0.  It  may  hap- 
pen that  these  factors  are  not  all  different ;  for  instance, 
(a:  —  otj)  may  occur  once,  twice,  or  any  number  of  times  up 
to  n  times,  in  which  case  «j  is  said  to  be  a  double,  triple,  etc. 
root  of  the  equation.  The  equation  is  still  said  to  have  n 
roots.  How  a  double  root  forms  the  transition  between  two 
real  and  distinct  and  two  conjugate  imaginary  roots  has 
been  shown  in  77. 


THEORY   OF   EQUATIONS  267 

323.  Descartes'  Rule  of  Signs.  Concerning  the  equation 
obtained  by  placing  f(x)  =  0,  Descartes  gave  the  following 
rule  in  regard  to  the  roots :  The  number  of  positive  roots  of 
an  equation  cannot  he  greater  than  the  number  c  of  changes  of 
sign  in  passing  from  the  first  to  the  last  term^  and  the  number 
of  7iegative  roots  cannot  be  greater  than  the  number  c'  of  changes 
of  sign  inf(  —  x')=  0, 

When/(ir)  =  0  is  a  complete  equation,  that  is,  an  equation 
which  contains  terms  involving  all  powers  of  x  from  aP  to  x"' 
inclusive,  if  p  denote  the  number  of  permanences  of  sign, 
c  -{-  2^)  =  n,  p  =  e\  and  therefore  c  -{-  c'  =  n,  but  if  the  equation 
is  not  complete,  c'  is  not  necessarily  equal  to  p. 

When  imaginary  roots  exist,  it  is  often  possible  to  de- 
tect their  presence  by  Descartes'  rule  of  signs.  For  when 
c  -{-  c'  <n^  that  is,  when  the  number  of  positive  and  negative 
roots  together  is  less  than  n,  the  whole  number  of  roots  of  the 
equation,  then  n—  (^c  -\-  <?')  is  an  inferior  limit  of  the  number 
of  imaginary  roots. 

EXAMPLES 

1.  Show  that  the  equation  x^— 2x^-\-4x^-Sx—2  =  0 
cannot  have  more  than  three  positive  roots  and  one  negative 

root.     The  series  of  signs  is  H 1 — ,  in  which  there  are 

three  changes  from  +  to  —  or  from  —  to  +, hence  there  cannot 
be  more  than  three  positive  roots.     If  x  is  changed  into  —  x^ 

the  signs  become  -\ — f-  +  H ,  where  there  is  but  one  change 

and  hence  there  cannot  be  more  than  one  negative  root. 

2.  Show  that  the  equation  a;*  —  3  a;  +  1  =  0  has  at  least 
two  imaginary  roots. 

3.  Show  that  the  equation  2  a;* +  5  a:^ +  3  =  0  has  all  its 
roots  imaginary. 

4.  Find  the  upper  limit  of  the  number  of  positive  and 
of  negative  roots  in  the  equation  x^  —  Sx^-\-x— 1  =  0. 


268  COLLEGE   ALGEBRA 

324.  Complex  Roots  enter  Equations  in  Pairs.  If  a-{-  pi 
is  a  root  of  /(a;),  a  —  /3z  is  also  a  root.  Substituting 
a  -|_  pi  for  X  in  f(^x)  and  collecting  the  real  and  imaginary 
parts  separately,  it  takes  the  form  /(«  +  /3i)  =  A  -\-  Bi^ 
which  by  hypothesis  vanishes,  that  is  A  +  Bi  =  0,  therefore 
A  =  B  =  0.  Since  a  —  (Bi  may  be  obtained  from  a  4-  Pi  by 
changing  the  sign  of  ^,  it  is  seen  that  fQi  —  pi)  =  A  —  Bi=0, 
since   A    and   B   are    zero ;    therefore    a  —  pi   is   a  root  of 

By  a  precisely  similar  process  it  can  be  shown  that  if 
^-^  V^  is  a  root,  a  —  V/^  is  also  a  root  of  /(a;)  =  0;  that  is, 
binomial  quadratic  surds  enter  equations  in  pairs  as  conjugates. 

Example.  The  equation  rr*  —  4a;^  +  4rr  —  1=0  has  2  +  V3 
for  one  root.     Find  the  other  roots. 


325.  Theorems.  If  a  and  h  are  tivo  numbers  of  which  b  is 
the  greater,  and  f(a)  and  f(b)  have  the  same  or  opposite  signs, 
then  an  even  or  an  odd  number  of  real  roots  of  f(x)  =  0  lies 
between  a  and  b. 

For  convenience  of  statement  zero  has  been  included 
among  the  even  numbers,  for  it  is  apparent  that  there  may 
be  no  real  root  when /(a)  and /(J)  have  the  same  sign. 

Let  ce^,  «2'  "31  ••>  ^r  ^^6  ^11  ^1^6  ^Gal  roots  of  f{x)  which  lie 
between  a  and  b.     Then 

f(x)  =(x-  a^)(x  -  «2)  •••  (^  -  f^,>)F(x), 

where  F(^a)  and  F(b')  have  the  same  sign  ;  for  if  they  had 
different  signs,  there  would  be  a  root  of  F(^x)  and  therefore 
another  root  of  fQc)  besides  a^,  a^,  •••,  a^  lying  between  a 
and  6,  which  is  contrary  to  hypothesis. 


THEORY   OF   EQUATIONS 


269 


Substituting  a  and  h  for  x^  we  have 


/(a)  =  (a-  a^{a  -  a^)  .-•  (a-  Ur^F^a). 


0) 


/(^)  =  (^-«i)(^-«2)  -  Cb-ar)Fiib). 


In  /(«)  the  factors  (a  —  «j), 
•  ••,  (a  —  a,,)  are  all  negative, 
while  in  f(b)  the  factors 
(6  — ccj),  •••,  (6  —  «,,)  are  all 
positive ;  if  /(a)  and  /(^) 
have  the  same  or  opposite 
signs,  the  products  (a— ctj).-- 
(«  —  «,.)  and  (^  —  ctj)  ••• 
(5  —  a^)  have  the  same  or 
opposite  signs ;  but  the 
product  (5  —  «^)  •••  (^  —  «,.) 
is  always  positive,  therefore 
the  product  (a  —  «^)  ••• 
(a  — «^)  is  positive  or  negative 
odd.     Therefore  when  /(a)  and 


(2) 


Fig.  30. 


Fig.  29. 

according  as  r  is  even  or 
/(^)  have  the  same  sign, 
the  number  of  real  roots  r 
lying  between  a  and  b  is 
even  (or  zero),  and  when 
/(a)  and  f  (by  have  opposite 
signs,  the  number  of  real 
roots  lying  between  a  and 
h  is  odd. 

Conversely,  if  an  even 
nu7nber  of  real  roots  of  f\x) 
lies  between  a  and  ^,  /(a) 
and  f(b)  have  the  same 
sie/ns,  and  if  an  odd  number 


270  COLLEGE   ALGEBRA 

of  real  roots  lies  hetween  a  and  5,  /(«)  and  f(h)  have  op- 
posite signs,  as  can  be  seen  from  the  expressions  (1)  and 
(2)  for  f(a}  and  f{b).  Graphically,  we  see,  if  there  is  a 
change  of  sign,  the  curve  representing  f(x^  must  cross 
the  axis  of  x  an  odd  number  of  times,  as  in  figure  29 ;  if 
there  is  no  change,  an  even  number  of  times,  as  in  figure  30. 

EXAMPLES 

1.  Prove  that   x^-\-S  a^—60  x^-\-2  x-^l  =  0  has  a  positive 
root  between  0  and  1,  and  a  negative  root  between  0  and  —  1. 

2.  Show  that  the  equation  x^  —  S  x^  -\-  x  —1  =  0  has  at  least 
one  positive  root  between  1  and  2. 

326.    Since 
f(x)  =  a^x""  +  «^a:''~i  + h  «„  =  a^(^x  —  a^ (x—  a^'-'(x—  «„), 

•^  ^x^^-'hx''-^  +  ...  +  ^  =  (:i;  _  «^)(^  _  ,,^)  ...  (^  _  ^^), 

we  have,  using  p^  for    ~^, 

% 

P\  =-(«i  +  «2  +  •••  +  0=-2«i, 


P2  =  (V2  +  ^^i«3  H 1-  ««-!««)  =  ^«i 


2' 


Fs  =  ~  ^"i 


«oO£ 


2^31 


Pr=  (-l)'"2«i«2 


i^«=(-l)''-«l«2 


a. 


For,  to  get  the  coefficient  of  a:"~^  in  the  product,  x  must  be 
selected  from  (^n  —  r)  factors  and  as  from  the  remaining  r 
factors  in  all  possible  ways. 


THEORY   OF   EQUATIONS  271 

327.  Cube  Roots  of  Unity.  To  find  the  roots  of  x^  =  1,  we 
have  Q(^  —  \  =  (x—  V)Qx^  +  a;  +  1)  =  0  ;  the  roots  are  therefore 
x=l  and  a:=  — |±|V  — 8.  If  we  denote  the  imaginary 
roots  by  &>  and  w',  it  is  easily  seen  that  (o'  =  (o^,  o)  =  w'^,  and 
(o(o'  =  1,  from  which  we  see  that  the  three  cube  roots  of  unity 
can  be  expressed  as  1,  o),  w^  and  that  1  -{-  co  -{-  (o^=0. 

EXAMPLES 


1.  x^-\-7/^=(^x-\-^^(^0)X  +  co'^^)((o'^x-\-(O7/^.     0)=  — -|-4- JV  — 8. 

2.  x^  —  y"^  —  {x  —  y~){wx  —  co^y^iccP'x  —  (oy). 

3.  {x-{-  (Dy  -\-  ccr'z)(x-{-  ccP'y-\-  coz)  =x^-\-y'^-\-  z^—  xy  —  yz  —  zx. 

4.  (x-\-y-\-  z')(^x  +  Qx^y  +  coz^  (^x  +  coy-\- «%)  =  a^ -\- y^ -{- z^ 

, .  —  3  xyz. 

5.  (l-ft))3=_3ft)(l-w)  =  -8V^=^. 

6.  (l-a))2  =  -3a). 

7.  li  x-\-  y  =  u^,  (OX  +  M^y  =  ti^,  ap'x  +  &)?/  =  7/3,  then 
{u^  -  u^)(u^^  -  '^3)0^3  -  u^)  =  -  3V-  3  (x^  -  ?/3). 

SYMMETRIC   FUNCTIONS 

328.  Definitions.  The  relations  between  the  coefficients 
and  the  roots  given  in  326  enable  us  to  express  certain 
functions  of  the  roots  in  terms  of  the  coefficients  without 
knowing  the  values  of  the  individual  roots. 

A  function  is  symmetric  with  respect  to  its  variables  when 
the  interchange  of  any  two  of  them  leaves  its  form  unaltered. 
Thus  X  +  y  -{-z^  2  x^  -{-  2  y^  -\-  2  z"^  -{-  S  xy  -\-  S  yz  -{-  S  zx  are  sym- 
metric and  are  denoted  by  Sa:,  2  ^x^  +  3  ^xy. 

The  functions  of  the  roots  contained  in  the  relations  of 
326  are  known  as  the  fundamental  symmetric  functions.  By 
means  of  these  relations  every  symmetric  function  can  be 
expressed  in  terms  of  the  coefficients. 


272  COLLEGE   ALGEBRA 

If  a,  /3,  7  be  the  roots  of  x^  -\-px^  -\-  qx  -\-  r  =  0^  we  may  find 
the  symmetric  function  Sa^  as  follows  : 

^'''''''  («  +  /3  +  ./)2=«2+^  +  ^2  +  2«/3  +  2;e7  +  2  7«, 

we  have  '^a^=  (Xa)'^  —  2  Xaff  =  p^  —  2  q. 

To  find  Xa^/3  we  have 

(«yS  +  /37  +  7a)(«  +  yg  +  7)  =  2^2^  +  3  afiy. 

Therefore  Xa^^  =  —pq  4-  3  r. 

.  ^2  +  ^^2+2  2_^«2 

ap  P7  7« 

^  7(«2  +  ^2)^,,^^2_^^2)_^^(^2^^2>) 

«/37 
—  ;?gH-  8  r 

—  r 

EXAMPLES 
1.    For  the  cwhio,  x^ -{- px^ -\-qx-\- r=(i^  find  the  symmetric 
functions    ^a^ ;    ^a^/3^;   (« +  ^)(^  +  7)(ry  +  «)  ;  2  "^  "^  ^^  ; 


a 


j3      y      a      a      /3      y 

2.  For  the  biquadratic  a;^  4- Jt>a;3  +  ^.^2 -^  ^2:  +  s  =  0,  find 
2«2^2.   5;,,2^^. 

329.  Factoring  of  Symmetric  and  Related  Expressions. 
From  the  definition  of  a  symmetric  function  it  is  apparent 
that  the  sum,  difference,  product,  and  quotient  of  two  symmetric 
functions  are  themselves  symmetric. 


THEORY  OF   EQUATIONS  273 

The  function  (^a—hx)(h  —  cx)(^c  —  ax)  is  unchanged  if  a 
is  changed  into  6,  h  into  <?,  and  c  into  a,  whereas,  if  we  inter- 
change a  and  5,  its  form  is  changed.  Though  such  a  func- 
tion is  not  symmetric,  it  is  said  to  be  cy  do -symmetric. 

The  function  aQ)^  —  (P')  +  h(^c^  —  a^)  +  ^(a^  _  52^  jg  changed 
into  its  negative  when  any  two  of  the  letters  a,  5,  c  are  inter- 
changed.    Such  a  function  is  said  to  be  alternating. 

The  sum.,  difference.,  product.,  and  quotient  of  two  cyclo-sym- 
metric  functions  are  cy  do -symmetric  functions. 

The  sum  a7id  difference  of  two  alternating  functions  are  al- 
ternating functions,  and  the  prodiict  and  quotient  of  two  alter- 
nating functions  are  symmetric  functions,  the  same  variables 
being  involved  in  each  case. 

The  preceding  propositions,  together  with  the  factor  theo- 
rem, make  it  very  easy  to  factor  certain  symmetric  and 
alternating  functions. 

EXAMPLES 

1.    Factor    a(P  -  (P)+ b(^c^- a'^)-{-c(ia^ -b'^^. 

Using  the  factor  theorem,  first  trying  for  monomial  factors, 
by  putting  <x  =  0,  we  see  that  this  does  not  make  the  expres- 
sion vanish  and  hence  a  is  not  a  factor.  Cyclo-symmetry 
then  shows  us  that  neither  b  nor  c  are  factors.  Next  try 
for  binomial  factors  by  putting  a  =  b.  In  this  case  the 
expression  vanishes  and  therefore  a  —  6  is  a  factor,  and  cyclo- 
symmetry  shows  us  that  b  —  c  and  c  —  a  are  also  factors. 
The  expression  being  of  the  third  degree,  there  can  be  no 
other  literal  factor,  hence 

Since  this  is  an  identity,  we  must  liave 

ab''-=Nab'^, 
and  therefore  iV=  1. 


274  COLLEGE   ALGEBRA 

2.    Factor     a^ (h  —  c)  -\- h^  (^c  -  a)  +  e^  {a  -  h) . 

This  expression  is  a  homogeneous,  alternating  function  of  the 
fifth  degree.  As  before,  we  find  that  a—h^  b  —  c,  and  c  —  a 
are  factors,  and  therefore  (^a  —  h^(h  — c)(^c— a)^  which  is  an 
alternating  function,  is  a  factor,  therefore  the  other  factor, 
which  is  the  quotient  of  the  given  expression  by  this  prod- 
uct, must  be  a  homogeneous  symmetric  function  of  the 
second  degree.     Hence, 

a\h  -  (?)  +  h\c  -  a)  +  c* (a  -  5)  =  (a  -  h^(h  -  e)(^c  -  a)[iV(a2 

+  h'^  +  6'2)  +  M(iah  +  hc^-ca)']. 

Since  this  is  an  identity,  we  have,  equating  coefficients, 

a^h  =  -  Na% 

and  therefore  iV=  —  1. 

Similarly,    a%W-  a^'^M^  0, 

or  iV^=  M, 

The  coefficients  il!f  and  iV  might  have  been  found  by  giving 
special  values  to  a,  ^,  c.  Sometimes  it  is  more  convenient 
to  use  one  method  and  sometimes  the  other,  or  in  many  cases 
it  is  most  convenient  to  combine  the  two.  Thus,  after  hav- 
ing found  N  by  equating  coefficients,  we  might  find  M  by 
substituting  in  the  identity  0,  1,  —  1  for  a,  5,  c^  respectively, 
and  obtain 

(-  1)  +  (-!)  =  (-  1)(2)(-  1)[-  (1  +  1)  +  Mi-  1)], 
from  which  we  get  iltf  =  —  1. 

Hence, 
a^(h-c)  +^*(^-^)  -\-o^(a-h) 

=  —  {a  —  b){h  -  c)(^c  -  a'){a'^  -\-  b"^  -h  c^  +  ab  +  be  +  ca). 


THEORY  OF   EQUATIONS  275 

330.  To  form  the  equation  whose  roots  are  the  roots  of  the 
equatio7i  f(x)  =  0  each  multiplied  hy  a  constant  factor  A,  we 
have,  if  a  be  any  root  of  the  equation  f(x)  =  0,  yg  =  ha  as  the 

B 
corresponding  root  of  the  sought  equation,  whence  a  =  - ,  and 

therefore,  since /(«)  =  0,  we  have/[y  1  =  0,  and  so  for  every 
root,  and  hence  if  y  represents  the  unknown  of  the  sought 
equation,  it  is  fi'^\  =  0.     lif(x)  is  the  polynomial 

a^x"  +  «i2:""i  _|_  ...  4-  ^^^ 
we  have  /?/V  ,       fv\~'^  ,  ,  a 

or  multiplying  by  h'\  the  equation  in  y  becomes 

a^yn  +  A^i^^-i  +  h\y''-^  +  ■"  +  h"a,,  =  0, 

in  which  it  is  seen  that  the  coefficients  are  formed  by  multi- 
plying aj.  by  h''  for  all  values  of  r  from  0  to  n. 

By  this  means  any  equation  with  fractional  coefficients  can 
be  transformed  into  one  in  which  a^  is  unity  and  all  other 
coefficients  are  integral.  Thus  if  we  multiply  the  roots  of 
theequation       ^4  +  j^  +  |^a  + |^+ 3  =  0 

by  30,  which  is  the  least  common  multiple  of  the  denomina- 
tors of  the  coefficients,  we  have 

x^-^lb3^-{-  300  x^  +  10800  X  +  2430000  =  0. 

Sometimes  a  smaller  number  than  the  least  common  multi- 
ple of  the  denominators  will  serve  to  free  the  equation  from 
fractional  coefficients. 

If  in  the  transformation  of  330  we  make  A  =  —  1,  the 
resulting  equation  becomes  one  whose  roots  are  the  negatives 
of  those  of  the  given  equation. 


276  COLLEGE   ALGEBRA 

331.     To  find  an  equation  whose  roots  are  the  reciprocals  of 
the  roots  of  the  given  equation^  we  have 


/(^)  =/(-)  =  0, 


or  ?a  +  -^+  ...  +a„=0, 

whence  by  multiplying  by  y^  we  have 

and  we  see  that  the  coefficients  of  the  resulting  equation  are 
formed  from  those  of  the  given  equation  by  changing  a^  into 
a„_^  for  all  values  of  r  from  0  to  n. 

Example.    Find  the  equation  whose  roots  are  the  recip- 
rocals of  the  roots  of  the  equation 

332.  To  form  an  equation  whose  roots  are  the  roots  of  the 
given  equation  diminished  by  A,  we  have,  if  ?/  be  a  root  of  the 
proposed  equation,  y  —  x  —  h^  and  therefore  x  =  y  -\-h^  hence 
f(x)  —fiy  +  A)  =  0  is  the  required  equation  in  y.  The  iden- 
tity /  (a;)  =/ (y  +  70  gives 

or  \i  y  be  replaced  by  its  equal  x  —  7i,  we  have 

a^x^+a^x''-'-^-  ...  +a,,  =  AQ{x-h}n-^A^(x-hy-'^+  ... 

■i-A^_^{x-h)+Ar,. 

If  we  divide  both  sides  of  this  identity  by  x  —  h,  the  remain- 
der will  be  the  value  of  the  coefficient  A„.     If  we  divide  the 


THEORY   OF   EQUATIONS  277 

resulting  quotient  by  x  —  h,  tlie  next  remainder  will  be  A,^_^, 
and  by  continuing  this  process  all  the  ^'s  may  be  found.  It 
will  be  observed  from  this  that  to  obtain  the  coefficients 
in  the  required  equation  we  have  but  to  make  repeated 
applications  of  the  remainder  theorem. 

333.    If  Q  is  the  quotient  and  R  the  remainder  (indepen- 
dent of  x)  when /(a;)  is  divided  hj  x—h^  we  have 

where  .§=  ^Qa^^-^-f- V^'^  +  •••  +  ^.-i- 

Equating  coefficients  of  like  powers  of  x  on  both  sides  of  the 
equation  we  have  the  following  relations, 

or,  as  they  may  be  written, 

bQ  =  aQ,  b^  =  hbQ  +  a^,  b^  =  hb^-\-  a^,  •••,  5^=  hb^.^-\-a^,  •••, 

R  =  hb^_^-{-a„, 

from  which  we  see  that  the  work  may  be  arranged  as  follows  : 

0  1  2    * '  *  ^ 

hb^     Jib^  •••  hbn-i 


h       ^1        h         ^ 


The  coefficients  of /(a;)  are  written  in  a  row,  zero  coefficients 
being  supplied  where  necessary.  The  coefficient  a^  is  brought 
down  as  b^.  This  is  multiplied  by  h  and  added  to  a^  giving 
by     This  in  turn  is  multijDlied  by  h  and  added  to  a^,  giving 


^2,  and  so  on. 


It  Avill  be  observed  that  this  process  gives  both  the  coeffi- 
cients in  the  quotient  and  the  remainder  when /(a:)  is  divided 
by  a:—  h.  If  the  remainder  is  zero, /(a;)  is  exactly  divisible 
by  a:  —  A,  or  has  x  —  h  as  ^  factor. 


278  COLLEGE   ALGEBRA 

EXAMPLES 
1.    To  diminish  the  roots  of  the  equation 

by  2,  we  have 


1 

-3 

2 

5 

2 

-2 

0 

1 

-1 

0 

5 

2 

2 

1 

1 

2 

2 

Hence  the  required  equation  is 

x^-\-  Sx^  -{-  2x -{-  5=  0. 

To  increase  the  roots  of  an  equation  by  h  would  be  the  same 
as  to  diminish  them  by  —  h. 

2.    To  increase  the  roots  of  the  equation 
x^-\-Sx^-\-  2  2^  +  5=0, 

by  2,  we  have 


1 

3 

2 

5 

-2 

-2 

1 

1 

0 

5 

-2 

2 

1 

-1 

-2 

2 

1     -3 

and  hence  the  required  equation  is 

a^-Sx^  +  2x-^5  =  0, 
which  is,  as  it  shouki  be,  the  equation  of  Example  1. 


THEORY   OF  EQUATIONS  279 

334.  EXAMPLES 

Find  the  successive  derived  functions  of : 
1.   x^-2x^-{-Sx-l.  2.   2x'^-:i:^-h2x'^-\-5x+l. 

3.  x^  —  ^  x^  -\-  7?'  —  5. 

4.  \if(x)  =  2:3  +  3  :r2  -  7  :r  +  2,  find/(2:  +  Ji). 

5.  \if(x)  =x^-2  x^  +  a;  +  1,  find/(2:  -  2). 

6.  Show  that  the  equation  x^  -\-  Z  x^  -\-l  x  =  ^  has  no  real 
root  except  zero. 

7.  The   equation  x^  —  :j^  —  ^  x^  -\-W  x  -\-  b  =  ^  has  a  root 
a;  =  2  +  V—  1.     Find  the  remaining  roots. 

8.  One  root  of  the  equation  a^*  —  6  a:;^  +  13  a;^  —  8  2:  —  6  =  0 
is  1  +  V2.     Find  the  remaining  roots. 

Find  for  the  biquadratic  a;*  +  'p:i^  +  qx^  +  ra;  +  s  =  0  : 

9.  2«2/37.  10.  2«3/3.  11.   2a*. 

Factor : 

12.  (^  -  2)5  +  (2  -  2^)5  +  (2;  -  ?/)5. 

13.  a5(^  — c) +^^(^  — «) +  <?^(rt  — ^). 

14.  x\y^  -  ^2)  +  if(z^  -  x^)  +  2;*(a;2  -  j/2). 

15.  (a;  +  ^  H- 2;)^  —  a:^  —  ^3  —  2^. 

16.  {a-\-h-\-  cy  —{h-\-  cY-  (c  +  ay  -  (a  +  5)*  +  a*  +  ^^  +  c^- 

17.  Transform  the  equation  x^-\-^a^  — ^x^-\-^x  +  l  =  0 
into  an  equation  in  which  the  coefticient  of  a;*  is  unity,  and 
all  the  remaining  coefficients  are  integers. 

18.  Transform  3  a:*  —  |  a:^  ^  7  ^2  _  1  ^  ^  2  =  0,  (1)  into  an 
equation  with  integral  coefficients,  (2)  into  an  equation  in 
which  a^  =  1,  and  the  remaining  a's  are  integers. 

19.  Find  the  equation  whose  roots  are  the  reciprocals  of 
the  roots  of  the  equation  2a:5— 3  a:*  —  4a;^H-a:2_52;_j_7-_o. 


280  COLLEGE   ALGEBRA 

Factor  and  solve  the  following  equations  by  the  method 
of  333  : 

20.  a;3-9a:2+26:r-24  =  0,  thus 

1  _  9  +  26  -  24  [2 

2-14     24 
1-7      12       0 

Hence  x—2  is  a  factor,  the  quotient  or  other  factor  is 
x^—lx-\-12;  proceeding  with  this  as  before  we  have 

1_74-12L3 

3-12 
1-4       0 

We  see  that  a;  —  3  is  a  second  factor,  and  the  quotient  or 
remaining  factor  is  a:  —  4.     Hence,  as  in  84,  we  have 

x^-9x^  +  2Qx-24:  =  (x-2)(ix-S)(x-4:), 

Hence  the  roots  of 

^3_9^2_f_  26a;- 24  =  0,  are  a;  =  2,  a;=  3,  :r=  4. 

21.  x^-Sa^-10x-^24:  =  0. 

22.  x^  ~  x'^ —  4:x  — 6  =  0. 

23.  a;*-29a;2  +  100  =  0. 

24.  x^-4a^-Sia^-\-16x-{-105  =  0, 

25.  Diminish  the  roots  of  the  equation 

a:3  _  2  a:2  +  5  a;  -  3  =  0,  by  3. 

26.  Increase  the  roots  of  the  equation 

2  a:3  +  4  a:2  -  3  a:  +  5  =  0,  by  2 . 

335.  The  Geometric  Interpretation  of  f'^Jr').  Let  P  andP' 
be  two  points  on  tlie  curve  j/  =f(x)^  whose  abscissas  are  x 
and  a:  +  A,  respectively. 


THEORY  OF   EQUATIONS 
Y 


281 


Fig.  31 


Then  we  have 


MP' 


h 


that  is, 


h 


=  tan  (/),  (317) 


or         /W  +  A  /'^(^)  +  A////(^)  +  ...  =  tan  0. 


2! 


3!' 


Now  as   h  =  0,  P'  =  P,  and  PP'  approaches  the  tangent 
PT  to  the  curve  at  P,  (^  =  r,  and  we  have 

f'(x)  =  tan  T.      (315  and  theory  of  limits) 

That  is,  the  first  derived  function  represents  tlie  slope  or  the 
tangent  of  the  angle  ichlch  the  tangent  Ihie  makes  with p)ositLve 
direction  of  the  axis  of  x.  When /'(.r)  is  positive, /(a:)  is 
increasing,  and  conversely.  When /'(a;)  is  negative, /(a;)  is 
decreasing,  and  conversely. 


282 


COLLEGE   ALGEBRA 


EXAMPLES 

1.  Find  the  slope  of  the  tangent  to  the  curve  y  =  x^  —  ^  x^ 
—  1,  at  the  point  (2,  —  5).  Here  f(x^  is  x^  —  Zx^  —  1  and 
f'(x)  =  3  a;2  —  6  2^,  which  for  the  point  (2,  —  5)  is  zero. 

2.  Find  the  slope  of  tangent  to  the  same  curve  as  in  the 
last  example  at  the  points  (3,  —  1),  (0,  —  1),  (1,  —  3). 

336.  Rolle's  Theorem.  Between  two  values^  a  and  h  (h  >a), 
of  X  for  which  f\x^  vanishes^  there  is  at  least  one  value  of  x  for 
which  f\x)  vanishes.  For  since /(a;)  is  continuous  between 
a  and  5,  it  must  first  increase  and  then  decrease,  or  first 
decrease  and  then  increase,  at  least  once,  as  x  passes  from  a 
to  h ;  that  is,  /'  (x)  must  change  from  positive  to  negative  or 
from  negative  to  positive  and  therefore  must  pass  through 
zero,  once  at  least,  as  x  passes  from  a  to  b. 

It  is  to  be  observed  that  we  have  assumed  in  this  proof 
that /(:c)  and /'(a;)  are  continuous  and  single-valued.  The 
accompanying  figure  illustrates  the  proposition. 


>X 


Fig.  32. 


THEORY   OF   EQUATIONS  283 

337.  From  the  preceding  theorem  it  follows  that  between 
two  consecutive  roots  of  f  (x}  =  0  there  may  he  no  real  root  of 
f{x)  =  0  and  there  cannot  be  more  than  one  such  root. 

338.  Theorem.  If  (x  —  ay  is  a  factor  of  f(x)^  then 
(x—ay~^  is  afactoroff'{x). 

We  have 
/W  ^fia+ix-a)^  ^fCa}  +  (:r-  «)/(«)  +  - 

^    Cx-ay-\p-i\a)   ^    (x-ayf^^^(a')   ^ 

(r  —  1)  I  r  !  * 

But  by  hypothesis  f(x)  =  (x  —  ayF(x^),  and  since  these 
values  of  f(x)  are  identically  equal,  the  coefficients  of  the 
corresponding  powers  of  (x  —  a)  must  be  equal,  and  there- 
fore/(«)=/(«)=  •••  =/'-i^(a)  =  0.  That  is,  a;- « is  a  fac- 
tor not  only  oif(x)  but  of  its  first  (r  —  1)  derived  functions. 

Similarly  we  have 

(r  -  2)  ! 

(x-  ay-lf^'\a)  _(a^-ay-if(>Xa) 

(r-1)!  O'-l)!  ' 

from  which  we  see  that  (^x  —  ay-^  is  a  factor  of /'(a;),  and  in 
the  same  way  we  see  that  for  the  other  derived  functions  we 
have  a  corresponding  result  giving  us  the  theorem.  If  a 
occurs  r  times  as  a  root  of  f(^x')=0,  it  ivill  occur  r—1  times 
as  a  root  off'(x)=  0,  r  —  2  times  as  a  root  of  f"  (x')  =  0,  and 
so  on  to  once  as  a  root  off''~'^\x^  =  0. 

The  development  oif(x')  shows  conversely  that  if 

/(a;)  contains  (x  —  a)  r  times  as  a  factor ;  that  is^  a  occurs  r 
times  as  a  root  of  f{x^  =  0. 


284  COLLEGE   ALGEBRA 

If/(x)  =  Ohas  a  root  repeated  r  times,  that  is,  a  multiple 
root,  it  can  be  found  by  determining  the  highest  common 
factor  oi  f{x)  and /'(a;). 

Example.      Find  the  multiple  root  of  the  equation 
Here  /(^)  =x'^  —  x^—%7?'-{-bx—^^ 

f^\x)  =  1\x-^, 

from  which  we  see  that  for  a;  =  1,  /(:r),  f'(x)^  and /''(a;) 
vanish,  and /(a:)  contains  (x—V)  three  times, /^ (a;)  twice, 
and /''(a;)  once  as  a  factor.     In  fact,  we  have 

/(^)=(:,_  1)3(^+2), 

/(a:)  =  (.T-l)2(4a;  +  5), 

f"{x)  =  (x-l)(12x  +  Q). 

339.  Theorem.  As  x  passes  through  a  root  a  of  f(x)  =  0, 
f(x)  and  f  {x)  have  unlike  signs  for  a  value  of  x  sufficiently 
near  and  less  than  a  and  like  signs  for  a  value  of  x  sufficiently 
near  and  greater  than  a. 

We  have     /(«  +  li)  =  hf  („)  + 1!/"  („)+.. ., 

/'(«+ A)  =/'(«) +  ¥"(«)+  -, 

and  since  the  first  term  of  the  right  member  determines  the 
sign  in  each  development  when  h  is  sufficiently  small,  we  see 
that  /(«  +  A)  and  /'(a  +  A)  have  opposite  signs  when  h  is 
negative  and  like  signs  when  h  is  positive. 

This  theorem  is  still  true  if  «  is  a  multiple  root  oif(x)  —  0. 
When  a  is  not  a  multiple  root  of  the  equation,  the  truth  of 
the  theorem  is  easily  seen  from  the  accompanying  figure. 


THEORY   OF   EQUATIONS 


285 


>x 


Fig.  33. 


Example.     Using  the  equation  of  the  example  of   the 
preceding  article,  we  have 

fix)  =  {x-  Vf(x  +  2), 
/'(a^)  =  (:K-l)2(4x  +  .5), 

and  we  see  that  for  a  value  of  a:  a  little  less  than  1,  f{x)  is 
negative  and /'(a:)  is  positive,  while  for  a  value  of  a:  a  little 
greater  than  1,  both  are  positive. 

340.    To  transform  a  given  equation  into  another  having  one 
term  less.     Let  us  write  for  convenience /(a;)  in  the  form 


/(a:)  =  a^x^'  +  na^x'^--^  +  '-^-%y^«2^"'^  +•••+«„. 


Then 


/(y  +  h)  =  r„y"  +  nv,r-'  +  ^^^1=^  r^"-^  +  -  +  K 


286  COLLEGE   ALGEBRA 

where     FQ  =  aQ, 

Fj  =  a^h  +  a^, 

F2  =  «Q/i2  +  2  a^li  +  «2' 

F;  =  ajf  +  mi7i'-i  +  ^^"^""^^  ajf^  +  •  •  •  +  ^.. 

^  1 

If  in  this  expression  for  /(?/  +  Ti)  we  wish  to  have  any 
one  of  the  terms,  say  (^)FJ-^""%  vanish,  it  is  only  necessary  to 

use  for  h  one  of  the  values  for  which  V,.  vanishes.  Thus  if 
we  wish  to  transform  the  cubic  a^x^  +  3  a-^x^  +  3  a<yX  +  a^  =  0, 
into  one  lacking  the  second  term,  we  have  J\  =  aji  +  a^,  or 

h= i,  so  that  the  cubic  in  y  becomes 

< 

«o2/'  +  -^-^-s — —y^^^^ ^ ^  =  0. 

or  transforming  it  into  another  whose  roots  are  the  roots  of 
this  equation  multiplied  by  ^q,  we  have 

Substituting  in  this  last  equation, 

z  =  a^y  —  a^x  +  a^,  11=  ciQd^  ~  ^ii 

and  G  =  «o^<^3  —  3  a^a-^a^  +  2  a-^^ 

we  have  2^  +  3  iT^  +  (7=0. 

Example.     Transform  the  equation  x^-{-2x^-\-Zx—2  =  0 
into  one  which  lacks  the  second  term. 


THEORY  OF   EQUATIONS  287 

341.    Solution  of  the  Cubic. 

z^-{-ZHz+  a  =  o. 
Let  z  =  ^/p  +  V^, 

then  2-^=^  +  ^4-3  ^pq  (  Vp  +  "v/^)» 

or  2^—3  Vp  V^  z  —  Qj-{-  q)  =  0. 

As  this  equation  is  to  be  identical  with 

z^-hSHz+  a  =  o, 

we  have,  equating  coefficients, 

s/p  -y/q  z=  —  H^  or  pq  =  —  H^^ 
p-\-q=  —  G-. 

Since  we  have  both  the  sum  and  the  product  of  p  and  q^ 
the  quadratic  equation  of  which  they  are  the  roots  is 

This  quadratic  is  called  the  reducing  quadratic  of  the 
cubic.     The  values  of  p  and  q  are  therefore 

p  = 

3-3-^  2 

Since  -^p  ^q  =  —  H^ 

3/-  H 

This  relation  between  Vjt?  and  ^q  determines  that  cube 
root  of  5',  which  is  to  be  associated  with  a  given  cube  root  of 
p.     We  thus  see  that  the  apparent  nine  values  of  Vp  +  v  g 


-  a 

+  V6^2 

+  4^3 

2 

-a- 

-V(7„ 

+  4^3 

288  COLLEGE   ALGEBRA 

are  limited  to  three,  as  they  should  be  to  make  the  substitu- 
tion legitimate. 

For  the  three  values  of  ^,  we  have 

h  =  S"i  +  «i  =  ^i?  4-  Vq  =Vp  —  -^— , 

h  =  S"2  +  «i  =  ^^P  +  co^Vq  =  co^p —^  =  (oVp  —  ^^, 

2!g  =  a^wg  H-  a^  =  o)^^^  +  wVq  —  co^Vp ^^-j^  =  co^^p g-^, 

where  a^,  0^2,  Wg,  are  the  roots  of  the  original  cubic  in  x. 
This  solution,  though  theoretically  correct,  is  not  in  a  form 
for  practical  use  when  all  the  roots  are  real ;  it  can  be  put 
into  such  a  form  by  means  of  trigonometry,  or,  better,  the 
solution  can  be  obtained  by  the  methods  of  316-318. 
342.    From  the  values  of  2^,  z^-,  %,  we  have 

By  multiplication  we  get 

V("i  -  "2)  ("2  -  «3)('^3  -  «i)  =  -  ^^-  ^^ip-q) 


=  _  3  V  -  3  V  (^2  +  4  ^3,      (yide  Exs.  2  and  5,  296) 
therefore 

This  relation  furnishes  a  means  of  determining  the  nature 
of  the  roots  of  our  cubic. 

If  the  roots  of  the  cubic  are  all  real,  the  differences  are  all 
real,  the  left-hand  member  of  the  relation  is  positive,  and 
therefor<^^^2_j_4^3  jg  negative. 


THEORY   OF   EQUATIONS  289 

If  the  roots  are  not  all  real,  two  of  them  are  complex  and 

the  three  have  the  form 

a^  =  a  -{■  bi, 

a^  =  a  —  M, 

and  therefore  «j  —  a.^  =  2  bi, 

a^  —  a^  =  a  —  c  -\-  b% 

a^  —  ci^  =  a  —  c  —  bi. 
Hence 

<(«!  -  «2)'(«2  -  «3)K«8-  «i)'=  -  4  b\<ia-  cy  +  b^y, 

and  therefore  (7^  +  4  JI^  is  positive. 

Conversely  if  ^  -f-  4  11^  is  negative,  the  roots  are  all  real, 
for  as  we  have  just  seen,  if  they  were  not,  (t^+  4  11^  would 
be  positive  ;  and  if  (^^  +  4  H^  is  positive,  one  root  is  real  and 
two  are  complex,  for  if  they  were  all  real,  G^-{-  4:  H^  would 
be  negative. 

If  (7^  +  4  H^  is  equal  to  zero,  two  roots  are  evidently 
equal,  and  conversely. 

If  (r  =  H=  0,  all  three  roots  are  equal,  for  in  this  case,  the 
equation  in  z  reduces  to  z^  =  0,  whence  z^  =  z^  =  z^,  or 

S^i  +  ^1  —  ^o^'^2  +  ^^1  ~  ^0^3  +  ^1  =  0,  or  «j  =  «2  =  ^^3  =  —  —  " 

343.  Discriminant.  It  may  be  found  that  (72  +  4  H^  con- 
tains the  factor  a^.  The  other  factor,  denoted  by  A,  is 
defined  as  the  discriminant^  that  is,  as  the  simplest  rational 
function  of  the  coefficients  whose  vanishing  expresses  the 
condition  for  equal  roots.     It  is  found  that 

A  =  ci^ci^  —  Qa^a^a^^fi^  4-  4  a^^a.^  +  4  a-^a^  —  3  a^a^. 


290  COLLEGE   ALGEBRA 

344.  The  Solution  of  the  Biquadratic.     The  equation 

a^x^  +  4  a^  4-  6  a^  +  4  a^  -|-  a^  =  0, 

(1 
can  by  the  substitution  x  =  y  —  -^  (340),  be  transformed  into 

a^(^a^a^  —  4  g^g^  +  3  ^2^)  —  ^(a^a^  —  a-^y^  _  r^ 
%^ 

or  multiplying  the  roots  of  this  by  a^,  (a;  =  «o^)'  ^^®  obtain 
2!4  +  6  ^^2  +  4  (7^  +  a^2j_  3  ^2  =  0, 

where  /=  a^a^  —  4  a^a.^  +  3  a^. 

345.  Assuming  2  =  Vp  +  V^y  +  Vr,  we  have 

z^  =  p  -\-  q -{-  7'  -\-  2(  Vpg  +  Vqr  +  -\^rp). 

Transposing  and  squaring  again,  we  get 
z^  —  2(p  -\-  q  +  r)^  +  C??  +  g  +  r)2=  \{jpq  -\-  qr -\-  rf) 

+  %^  'pqr(^'\/ f  +  V^  + Vr), 
or         2*  —  2  (p  +  3'  +  r)2;2  —  '^^pqr  z -\- {^p  -\-  q -\-  r)2 

—  4(pg  +  ^r  +  rp)  =  0. 

Since  this  equation  is  to  be  identical  with  that  of  344,  we 
have,  by  equating  coefficients, 

p  +  q-^r=^  —^  H, 

Vpqr  =  -—, 

Qp  +  q-^-rY—  4(pg'  -\-  qr  -{-  rp)  =  a^I—  3  H"^^ 
or  pq-{-  qr  +  rp=^  H^ ^ . 


THEORY  OF   EQUATIONS  291 

Hence  the  equation  wliose  roots  are  />,  g,  r,  is 

w3  +  3 Hii^  +Ur2_  <1\u -  ^  =  0, 

which  is  known  as  Eider  s  cubic.    By  adding  and  subtracting 
the  terms  H^  and  -*^— — ,    this  may  be  written 

•±  ■±  •± 

or         ^(u-\- H)^  —  a^I(ii  +  H)  +  aQ^a^a^a^  +  2  a^^g^g  —  a^^ 

Putting 

!<-  +  11=  a^v  and  ?/=  a^a^a^  -\-  2  a^a^a^  —  ^9'^  —  ^0^3^ "~  ^i^^4^ 
we  have  4  ^^^y^  —  J^^^ij  +  e/=  0, 

which  is   called   the  reducing  cubic  of   the  biquadratic.     If 
^1'  ^2'  ^3  "^^^  ^^^®  roots  of  this  equation,  we  have, 

r  =  a^  —  a^a^  +  ^o^^3* 

As  ^=  Vp  +  V^  + Vr,  it  might  seem  that  there  would  be 
eight  values  of  2,  but  as  in  the  case  of  the  cubic  these  are 
limited  to  four  by  the  relation 

VpVo'V?'  = ,   or  ^/^  = , 

so   that  when  jt>  and   q  are  chosen  r  is  determined  by  this 
relation. 


292  COLLEGE   ALGEBRA 

If  2j,  ^2,  z^,  z^  be  the  roots  of  the  equation  in  ^,  and  ^i,  a^-^  ^3, 
^4  the  roots  of  the  equation  in  x^  we  have 

z^  —  a^a^  +  «i  =  V^  +  Vg  +  Vr, 

;22  =  V2  +  ^1  —  ^i^  ~"  ^S"""  "^^' 
2g  =  a^Wg  4-  a^  =  _  Vp  +  V^  —  Vr, 

^4  =  ^0^4  +  «i  =  —  VJ9  —  V^'  +  Vr. 
STURM'S  THEOREM 

346.  Unequal  Roots.  If  the  roots  oif(x)  =  0  are  all  differ- 
ent, that  is,  if  /(a;)  and/'(:r)  have  no  common  factor  and  if 
/2(^)'/3(^)'  ">/»(^)  ^^'6  functions  obtained  like  the  remain- 
ders in  the  process  for  finding  the  highest  common  factor  of 
f^x}  and/'(a:)  except  that  in  this  case  the  sign  of  each  re- 
mainder is  changed  before  proceeding  to  the  next  division, 
we  have  the  following  theorem  due  to  Sturm  :  If  a  and  h  are 
any  two  real  numbers^  b>a,  the  excess  of  the  number  of  changes 
of  sign  in  the  series  of  functions  f{x'),  f'(x')^  f^(x~),  •••,/„ (2;), 
when  a  is  substituted  for  xover  the  number  when  b  is  substituted 
for  x^  is  the  number  of  real  roots  lying  between  a  and  b. 

It  is  obvious  that /,,(:r)  is  a  constant,  for  otherwise /(a;) 
and/'(:r)  would  have  a  common  factor. 

From  the  process  for  finding  the  functions  we  have  the 
following  relations  : 

/(^)  =  ^2/2  W -/sW 

/2(^)=^y3/3(^)-/4(^) 


fr-l(x)  =  qrfXx)  -fr+lix) 


THEORY   OF   EQUATIONS  293 

Here /(a;)  and /'(a;)  cannot  vanish  for  the  same  value  of  x^ 
that  is,  can  have  no  common  factor,  for  if  they  had,  f{x) 
would  contain  that  factor  at  least  twice  (338),  but  by  hy- 
pothesis this  is  not  the  case. 

No  two  consecutive  functions  of  the  series  can  have  a  com- 
mon factor,  for  if  they  had,  it  follows  from  the  foregoing 
relations  that  it  would  be  a  factor  of  all  the  functions  of  the 
series,  including  f{x)  and  /'(a;),  which,  as  we  have  seen,  is 
not  the  case. 

To  determine  the  loss  of  changes  of  sign  in  the  series  of 
functions  we  have  to  investigate  the  following  cases : 

1.  When  X  passes  through  a  root  of/(a;)  =  0. 

2.  When  x  passes  through  a  value  which  causes  one  or 
more  of  the  auxiliary  functions  f'(x)^  f^(x)^  •••  to  vanish, 
provided  that,  if  more  than  one  vanish,  no  two  of  those  which 
vanish  are  consecutive. 

First  Case.  If  a  be  a  real  root  of/(rr)=  0,  lying  between 
a  and  ^,  then  as  x  passes  through  a^fQx^  and/'(a;)  have  un- 
like signs  just  before  and  like  signs  just  after  the  passage 
(339).     Hence  one  change  of  sign  in  the  series  is  lost.   ^ 

Second  Case.  If  a  causes  /,.(^)  to  vanish,  that  is,  if 
/^(a)  =  0,  we  have/^_i(«)  =  — /r+i(«)-  ^^  ^^^  ^'^^^  values  of 
X  sufficiently  near  to  a  to  exclude  roots  of  /,.-i(a:)  =  0,  and 
/r+i(^)  =  ^'  we  see,  from  the  above  relation,  that  for  such 
values /,,_i(a;)  and/,.+i(a;)  have  signs  opposite  to  each  other, 
and  therefore  whether /^(a;)  changes  sign  or  not  as  x  passes 
through  «,  the  series  of  three  functions  /,._i(2:),/r(a:;),/r+i(^) 
presents  one  permanency  and  one  change  of  sign,  though  not 
necessarily  of  the  same  order  of  arrangement,  both  before  and 
after  x  passes  through  «. 

Thus  if  the  signs  of  fr-\(x)  and  fr+\(x)  are  —  and  -f-  re- 
spectively immediately  before  and  after,  and  if  fr(x)  is  — 


294  COLLEGE   ALGEBRA 

immediately  before  and  +  just  after  x  passes  through   the 

value  a,  we  have  before  the  passage  the  series  of  signs h , 

and  just  after,  the  series  — h  +,  or  one  permanency  and  one 
change  in  each  case.  If  /^C^)  does  not  change  sign,  that  is, 
if  a  occurs  an  even  number  of  times  as  a  root  of  fr(^x)  =  0, 
we  might  have,  for  example,  the  series  — h  +  both  before 
and  after  the  passage.  Hence  in  the  second  case  no  change 
of  sign  is  lost  or  gained  in  the  series  of  three  functions.  If 
others  of  the  auxiliary  functions  vanish,  the  same  thing  is  true 
for  each  of  the  corresponding  series  of  three  functions. 

Thus  we  have  proved  that  as  x  passes  through  a  real  root 
oif(x)  =  0  one  change  of  sign  is  lost  in  the  series  of  func- 
tions/(a;), /'(a;),  •  ••,/„ (a;)  and  in  no  other  case  is  a  change  of 
sign  lost  or  gained.  Hence  the  number  of  changes  of  sign 
lost  as  X  passes  from  a  to  b  is  the  number  of  real  roots  of  /(a;) 
=  0,  lying  between  a  and  h. 

The  loss  of  changes  of  sign  between  /(:r)  and  /'  (a;)  hap- 
pens by  means  of  the  rearrangement  of  the  signs  of  the 
series  as  x  passes  from  root  to  root. 

347.  Equal  Roots.  If  a  occurs  r  times  as  a  root  oi  f(x)  = 
0,  that  is,  if  (re  —  ccy  is  a  factor  of  /(r?^),  {x  —  a)'-i  is  a  factor 
oif'(x)  (338),  and  by  the  highest  common  factor  theory,  it 
will  be  a  factor  of  each  of  the  functions /g (a;), /g (a;),  ■'■^fj^(x)^ 
where /^.(a:)  is  that  remainder  which  exactly  divides  the  pre- 
ceding one  and  with  which  the  process  terminates.  Simi- 
larly, if  f(x)  contains  other  multiple  roots,  fk(^x)  contains 
them  each  to  a  degree  one  less  than  they  are  contained  in 
f{x)*  Consequently,  if  we  divide  the  functions  fix)-, 
f'(x)^'"^fj^(x)  and  also  the  relations  existing  between  them 
(346)  by  fk(x)^  we  obtain  a  new  series  of  functions  F(x)^ 
F^(x)^  •••,  Fj^(x\  where  F(x)  contains  each  distinct  factor  of 
f(x)^  once  and  only  once,  and  a  new  set  of  relations.     It  is 


J 


THEORY   OF   EQUATIONS  295 

apparent  that  the  terms  in  this  series  of  functions  will  have 
the  same  signs  or  signs  opposite  to  those  of  the  original  series 
according  as/;-.(^)  i^  positive  or  negative,  and  hence  v^ill  pre- 
sent the  same  changes  or  permanencies  as  the  original  series. 
Therefore  we  have  a  new  series  of  functions  possessing  the 
same  properties  with  respect  to  loss  or  gain  of  changes  of 
sign  as  the  functions  of  346,  and  hence  the  reasoning  and 
results  of  that  section  are  applicable  to  the  function  F(x).  . 
Hence  Sturm's  theorem  holds  for  real  multiple  roots,  count-/ 
ing  each  multiple  root  once. 

Example.  Determine  the  number  and  situation  of  the 
real  roots  of  the  equation  2^  —  3  a;^  +  5  a;  —  1  =  0. 

The  signs  of  the  terms  are  -\ 1 ,  whence  by  Descartes's 

rule  there  cannot  be  more  than  three  positive  roots.  Chang- 
ing X  into  —  ic,  the  signs  are  all  negative,  hence  there  can  be 
no  negative  root.  There  is  at  least  one  positive  root,  since 
the  absolute  term  is  negative.  To  determine  whether  all 
the  roots  are  positive  we  must  use  Sturm's  theorem.  The 
work  is  as  follows: 

f(x)  =  x^  —  ^  x^  +  b  X  —  \^ 
/(a;)  =  3  2:2 -6  a; +5. 

Dividing  3/(a;)  by/^(a;),  we  get  2:  —  1  as  quotient  and  4:x-\-2 
as  a  remainder.  Dividing  this  remainder  by  2  and  chang- 
ing its  signs,  we  have  —2  2;  —  1  for/2(2:).  Dividing  2/' (2;) 
by  f^Cx),  we  get  —  3  2:  +  15  as  quotient  and  35  as  a  remain- 
der, and  therefore /3(2:)  =  —  35. 

For  x=  —  OD  the  signs  of  the  functions  are h  H —  and 

for  x=  ao  they  are  +  H ,  therefore  as  one  change  of  sign 

is  lost  there  is  only  one  real  root.     For  2:  =  0  the  signs  are 

— I and  for  x=l  they  are  +  H ,  and  as  one  change 

of  siofn  is  lost  between  0  and  1  the  root  lies  in  that  interval. 


296  COLLEGE   ALGEBRA 

348.  Solution  of  Numerical  Equations.  The  real  roots  of 
numerical  equations  are  either  commensurable  or  incommen- 
surable, and  their  number  and  situation  may  be  determined 
by  the  use  of  Sturm's  theorem  and  the  principles  of  319-325. 
The  former  class  includes  integers  and  rational  fractions, 
and  since  every  equation  can  be  reduced  to  one  having  the 
coefficient  of  the  first  term  unity,  and  the  coefficients  of  its 
other  terms  integers,  the  problem  of  finding  the  rational 
roots  of  an  equation  reduces  itself  to  the  problem  of  finding 
the  rational  roots  of  an  equation  of  this  type. 

Let  the  equation  of  this  form  be 

x'^  +  a^a;^-!  +  a^x""-^  H h  ««  =  0 

where  the  a's  are  integers.  The  commensurable  roots  of  this 
equation  must  be  integers,  for,  if  not,  suppose  that  ^,  a  com- 

mensurable  fraction  reduced  to  its  lowest  terms,  is  a  root. 
Then 

or  multiplying  both  members  by  q^~'^  and  transposing, 
—  —  =  <2jjt?"i    +  a^p^  ^q-{- h  a,,q^~'^' 

The  right-hand  member  of  this  last  equation  is  an  integer, 
while  the  left-hand  member  is  not  an  integer,  since  p  and  q 
are  prime  to  each  other,  hence  the  supposition  that  our  equa- 
tion can  have  a  fractional  root  is  false,  and  hence  its  com- 
mensurable roots  must  be  integers. 

Since  a„  is  numerically  the  product  of  all  the  roots,  we 
may  find  the  commensurable  roots  by  using  the  integral 
factors  of  a^  according  to  the  method  of  333. 


THEORY   OF   EQUATIONS  297 

Example.  Find  the  commensurable  roots  of  4  a:*  —  8  o:^ 
4-  23  x^  —  4:0  x+  15  =  0.  Forming  an  equation  whose  roots  are 
the  roots  of  this  equation  multiplied  by  2,  we  have  (t/  =  2  x^ 

y  -  4  ?/3  +  23  ?/2  -  80  y  +  60  =  0. 
The  factors  to  be  tried  are  ±1,  ±2,  etc.     Dividing  by  ?/  — 1, 
we  have  1     _4       23-80         60 

1     -3     +20     _60 
1-3       20-60 

which  shows  that  1  is  a  root,  and  i/^  —  S  ^^  -^  20  t/  —  QO  is  the 
quotient.     If  we  divide  the  quotient  by  ^  —  2,  we  have 

1     _3       20-60 

2-2         36 

1     _1       18-21 

which  shows  that  2  is  not  a  root.  Similarly  we  may  find 
that  3  is  a  root,  and  that  the  other  factors  are  not  roots. 
Since  1  and  3  are  roots  of  the  equation  in  j/,  ^  and  |  are  the 
corresponding  roots  of  the  equation  in  x,  and  are  its  only 
commensurable  roots. 

349.  Horner's  Method.  When  it  is  found  that  a  positive 
root  lies  between  two  consecutive  integers,  we  may  form  an 
equation  whose  roots  are  the  roots  of  the  given  equation 
diminished  by  the  smaller  of  the  two  integers  which  is  the 
integral  portion  of  the  root  sought.  If  now  we  multiply  the 
roots  of  the  resulting  equation  by  ten,  we  can  again  find  two 
integers  between  zero  and  ten  such  that  a  root  of  this  equa- 
tion lies  between  them.  This  root  divided  by  ten  is  the 
remaining  portion  of  the  original  root,  and  therefore  the 
smaller  of  the  two  integers  is  the  next  digit  of  the  root.  By 
a  repetition  of  this  process  as  many  of  the  digits  of  the  root 
may  be  obtained  as  is  desired. 


298  COLLEGE   ALGEBRA 

Thus  given     f{x)  =  2^ -I'lx^ +?Ax -  b  =  0, 

We  find  tliat/(3)  is  positive  while /(4)  is  negative,  indicat- 
ing the  presence  of  at  least  one  real  root  lying  between  3 
and  4.     Sturm's  functions  are  as  follows  : 

f(x)  =  2^3  _  12  ^2  ^  31  ^  _  5^ 
/(^)=3a;2-24  2:+31, 

It  is  not  necessary  to  find  f^^x^  since  all  we  need  is  its  sign, 
which  can  be  determined  without  finding  the  function.  The 
sign  of  this  function  is  positive,  since  the  value  Jg^-  for  which 
/2(^)  vanishes  makes/'(a^)  negative, and  therefore y3(a7)  must 
be  positive  (346,  2). 

For  2:=  —  00,  tlie  series  of  functions  have  the  signs  — | h, 

and  for  a;  =  00,  the  signs  +  +  +  +  •  Hence  between  —  00  and 
-{-co  three  changes  of  sign  have  been  lost,  therefore  the 
equation  has  three  real  roots.     Again  for  x  =  3,  the  series  of 

functions  have  the  signs  H h  and  for  :c  =  4,  the  signs 

are f-  +,  that  is,  one  change  lias  been  lost,  and  therefore 

but  one  root  lies  between  3  and  4.  The  work  of  obtaining 
the  rest  of  the  root  to  four  decimals  is  as  follows : 

Diminishing  the  roots  by  3 

1  -12  31  -5 

3  -  27  12 


1 

-9 
3 

4                           T 

-18 

1 
1 

-6 

3 

-3 

-14 
Multiplying  the  roots  by  10  and 
then  diminishing  them  by  4 

THEORY   OF   EQUATIONS 


299 


1 

-30 
4 

-1400 
-104 

7000 
-6016 

1 

-26 
4 

-  1504 

-88 

984 

1 
1 

-22 

4 

-18 

- 1592 
Multiplying  th 
climinishin 

le  roots  by  10  and 
Lg  them  by  6 

1 

-180 
6 

-  159200 
-1044 

984000 
-  961464 

1 

-174 

6 

-  160244 
-1008 

22536 

1 

1 

-168 

6 

-162 

-161252 
Multiplying  the  roots  by  10  and 
diminishing  them  by  1 

1 

-1620 
1 

- 16125200 
-1619 

22536000 
-16126819 

1 

-1619 
1 

-16126819 
-1618 

6409181 

1 
1 

-1618 

1 

-  1617 

-16128437 
Multiplying  the  r 
diminishing 

oots  by  10  and 
them  by  3 

1 

-16170 
3 

-1612843700 

-  48501     - 

6409181000 

-  4838676603 

1 

- 16167 
3 

-  1612892201 
-  48492 

1570504397 

1 

- 16164 

-  1612940693 

1 
1 

3 
-16161 

1 

-  16161 

- 1612940693 

1570504397 

300  COLLEGE   ALGEBRA 

The  root  to  four  decimal  places  is  therefore  3.4613.  This 
method  of  obtaining  a  root  of  an  equation  is  known  as 
Horner's  method. 

350.  If  two  roots  are  nearly  equal,  that  is,  if  the  digits  are 
the  same  in  each  to  some  point  in  their  decimals,  we  proceed 
as  before  until  Ave  come  to  a  point  where  two  roots  are  found 
to  lie  between  two  different  consecutive  integers  between  zero 
and  ten,  when  they  will  begin  to  separate  and  each  can  be  cal- 
culated separately.  The  complete  solution  of  this  case,  how- 
ever, requires  more  detail  than  the  scope  of  this  work  admits. 

If  a  root  is  negative,  we  may  transform  the  equation  into 
one  whose  roots  are  the  negatives  of  those  of  the  given  equa- 
tion and  proceed  as  for  a  positive  root. 

351.  EXAMPLES 
Solve  the  equations: 

1.  x^-6x'^-hllx-6  =  0,  4.  x^-1x-6  =  0. 

2.  x^-2x'^-x-\-2==0,  5.  x^-^2x^-5x~6  =  0. 

3.  x^-2x^-5x-^Q  =  0.  6.  x^-7x-{-Q  =  0. 

7.  x^-5x^-(j4:x-i-U0  =  0. 

8.  x^-103^-\-S5x^-50x-{-24:  =  0. 

9.  x^-7x^-^bx^-\-Slx-S0  =  0. 

10.  a:4-19a;2  +  lla;  +  30-0. 

11.  x^-7x^-\-dx'^-7x-10  =  0. 

12.  x^-Ux^-^4:9x^-S6  =  0. 

Suggestion.     Put  x^  =  z. 

13.   2:6  —  52:^—2:2+5=0.  14.    2:^-2:6-64  2:3+64  =  0. 

15.  G,T3_8ia,2_p53^_30=0. 

Suggestion.  Transform  to  an  equation  having  the  coefficient  of  the 
highest  power  of  x  equal  to  unity. 


THEORY   OF   EQUATIONS  301 

16.  S0a^-llx^  +  59x-U  =  0. 

17.  iJx^-lr^-lS3^-{-Ux-h6  =  0. 

18.  x^  —  x^  —  9  x'^  —  3  X  +  2  =  0. 

19.  12  x^  +  44  x^  +  23  2;2  -  28  2:  +  5  =  0. 

20.  lSx^  +  S'^x^-2x^-7  =  0. 

21.  x^-7  x-]-o  =  0. 

Suggestion.     Apply   Sturm's  theorem  and  plot  the  curve.      Solve  by 
Horner's  Method. 

22.  a:3  +  5a;  +  3  =  0. 

23.  Find  the  positive  root  of  2:^  —  6  :r  —  13  =  0. 

24.  Solve  the  equation  x^  +  x^  -{- 1  =  0. 
Suggestion. 

x<i^x^+l=:0  or  x^  +  l-^—=x^  +  s(x  +  -\  +  —  -3(x-\--]  +  l 

x^  V       x)     x^        \       x) 

=  {x  +  -y  -?,lx  +  ~\  +  l=z^-^z+\=0. 

25.  Find  all  the  roots  of  the  cubic 

and  show  that  the  equation  ma}^  be  regarded  as  a  reduced 
form  of  x^  +  x^  -\-  x^  -\-  x^ -{■  x"^ -\-  x  +  1  =  0. 


CHAPTER  XIX 
MISCELLANEOUS  TOPICS 

MATHEMATICAL   INDUCTION 

352.  Some  further  work  on  mathematical  induction  is  here 
given  for  those  who  care  for  it.  In  169  mathematical  induc- 
tion has  already  been  used  to  prove  the  binomial  theorem. 

353.  Divisibility  of  x'^  ±y^  by  x  ±y.     We  know  by  trial 

that  x^  —  y^^  x^  —  y^^  are  divisible  by  x  —  y. 

Since  x^  —  y'^  =  x^  —  xy'^~'^  +  xy^~'^  —  y^ 

=  x(x''-^  -  /^-i)  +  y^'-^ix  -  ?/), 

we  see  that  \i  x  —  y  divides  a;""i  —  ?/^~i  it  also  divides  x^  —  y^^ 
for  any  value  of  n  —  1.  We  have  seen,  however,  that  a;^  —  y^ 
is  divisible  by  x  —  y^  therefore  x^  —  y^  is  divisible  by  x  —  y^ 
hence  x"  —  ?/^,  and  so  on  continually.  Hence  x"^  —  y'^  is  divis- 
ible by  X  —  y^n  being  any  positive  integer,  and  x  and  y  hav- 
ing any  values.     The  quotient  is 

^  =  x""-^  -I-  x^'-'^y  +  x^'-^y'^  H h  2:2/^-2  +  ?/"-i . 

^~y  (See  Ex.  10,  86) 

If  in  this  identity  we  change  y  into  —  y,  we  have 

^!^l!  =  a;'^-l  _  ^^-2^  ^  x^'-^yi  +  ...  _|_  (-l)'^-2.^:^»-2 

according  as  n  is  even  or  odd. 

Since  x"^  +  ^'^  =  2:"  —  ^"  +  2  ^^,  x"^  +  y'^  is  not  divisible  by 
x-y. 

302 


MISCELLANEOUS   TOPICS  303 

Thus  for  a  positive  integer  w, 

x—y  never  divides  a;" -f  ?/", 

X  —  y  always  divides  ^"  —  ?/% 

x-\-  y  divides  x^  —  y''^  when  n  is  even, 

x-\-  y  divides  x^  +  y'^  when  n  is  odd. 

The  student  may  prove  that 

x-\-  y  does  divide  x^  —  y^  when  n  is  odd,  and  that  x+  y 
does  not  divide  x'^  +  ?/^*  when  n  is  even. 

354.  Summation  of  Series.  The  method  of  mathematical 
induction  may  sometimes  be  applied  to  find  the  sum  of  a 
series. 

EXAMPLES 

1.  Find  the  sum  of  the  series  1,  3,  5,  etc. 

The  sum  for  one  term  is  1,  for  two  terms  is  4,  for  tliree 
terms  is  9,  that  is,     „  _  i    „  —  92   «  _  q2 

*1  —  ^1  ^2  —  ■"  '  *3  —  ^  * 

Let  n  be  the  greatest  value  for  which  and  for  all  lower 
values  of  which  we  know  the  law  Sn  =  n^  to  be  true.  Then 
since  ^^^  =  2n-l,  if«+i  =  2w  +  l, 

s,,^^  =  s„  +  W;,+i  =  w2  +  2  ^1  +  1  =  (?^  +  1)^- 
Hence  the  law  holds  for  a  value  one  more,  and  therefore 

for  any  positive  integral  value  of  n. 

2.  Prove  that  the  sum  of  the  series 

1.2  +  2-3  +  3.4+  •••  +</i  +  l)  is  l?iO^  +  l)(w  +  2). 

We  are  to  prove  here  that 

s„  =  iwOi  +  l)(n  +  2) 


304  COLLEGE   ALGEBRA 

The  reasoning  can  be  carried  out  as  before,  or  it  may  be 
obtained  by  indirect  reasoning  as  follows : 

si  =  1.2  =  i(l)(2)(3), 

«2  =  K2)(3)(4), 

S8  =  K3)(4)(5). 

If  the  law  here  observed  fails,  it  must  fail  for  a  first  time, 
say  for  the  (ti  +  l)th  case.  Then  by  hypothesis  it  does  not 
fail  for  the  nth,  and  hence 

and  therefore 

««+i  =  Sn  +  ^M+i  =  i  n(n  +  V)(n  -f-  2)  +  (?i  +  V)(n  4-  2) 

=  (n  +  1)(^  +  2)(i  7i  +  1)  =  IQn  +  l)(n  +  2)(n  +  3), 

hence  the  law  does  in  fact  hold  for  the  (n  +  1)  th  case  also, 
and  no  first  failure  does  occur,  and 

s„  =  i?i(w  +  l)(n  +  2) 
is  true  generally. 

355.    Prove 

>  (^i«i  +  A^a^  +  •••  +  A^a,^'^. 

Let  n  be  the  greatest  value  for  which  and  for  all  smaller  values 
this  inequality  has  been  shown  to  be  true.     Then,  since 

A         If,        2  _    /|         If,        2 

J        2^2i/12^        2-^<9J^J         ^ 
-^^n+i  ^^2      '         2  "'w+1    ^  "^  -'-*-2"'2      ^i+l^i+1' 

>4  2^2_L./12^  2-^9/1/yJ  /7 

(^A^^A;--^    ...    +A„2)(a^2_^^^^2+    ...   +^^2>) 

>  iA^a^  +  ^2^  +  •••  ^«^«)^' 


MISCELLANEOUS   TOPICS  305 

we  have,  adding  corresponding  members  of  the  equation  and 
of  the  inequalities,  and  combining, 

Hence  the  inequality  is  true  for  a  value  of  n  one  greater. 
But  when  n  is  2  it  is  easily  seen  to  be  true,  and  therefore  it 
is  true  when  n  is  3,  and  when  n  is  4,  and  so  on  generally  for 
any  positive  integral  value  of  n. 

356.  EXAMPLES 

Prove  by  mathematical  induction  : 
1.    1  .  3  .5  •••  2  7i-l<n'\ 


nJ  \        nJ        \  n      J      n\ 


3.    1.3-f2-4  +  3-5+---to7i  terms 

=  J^i(7i  +  l)(2n  +  7). 

8.    13  +  03  +  33  +  ...  +  n^  =  Mn±llJ. 

LIMITS 

357.  In  190  we  have  stated  four  elementary  theorems  of 
limits.  Before  we  can  satisfactorily  prove  them  it  is  neces- 
sary to  consider  briefly  the  theory  of  indefinitely  small  and 
indefinitely  great  numbers.* 

358.  Definitions.  An  indefinitely  small  number  is  a  variable 
whose  absolute  value  under  the  conditions  of  its  statement  may 
be  made  less  than  any  assigned  positive  number  however  small, 

*  These  are  the  infinitesimals  and  infinites  of  calculus. 

X 


306  COLLEGE   ALGEBRA 

Hence  we  see  that  its  limit  is  zero,  but  the  indefinitely  small 
number  itself  is  not  zero. 

An  indefinitely  great  numher  is  a  variable  ^vJiose  absolute 
value  under  the  conditions  of  its  statemefit  may  be  made  greater 
than  any  assigned  positive  number  however  great.  It  follows 
that  it  increases  without  limit,  but  is  not  itself  equal  to 
absolute  infinity,  i.e.  to  the  reciprocal  of  absolute  zero. 

359.  Theorem.  The  reciprocal  of  an  indefinitely  small  number 
is  indefinitely  great.,  and  the  reciprocal  of  an  indefinitely  great 
number  is  indefinitely  small. 

Proof.  Let  x  be  any  indefinitely  small  number  and  let  A 
be  any  assigned  positive  number.     Then  —  is  also  a  positive 

r^  1 

number  that  can  be  assiscned.     Therefore  x<  — ,  therefore 

1  1 .  .      .  ^ 

->  A^  therefore  -  is  indefinitely  great. 

XX 

Let  X  be  any  indefinitely  great  number.     Then  X>— , 

11 

therefore  —  <A,  therefore  —  is  an  indefinitely  small  number. 

X  X  -^ 

360.  Theorem.  The  product  of  an  indefinitely  small  number 
by  any  number  a  not  indefinitely  great  is  indefinitely  small. 

Proof.  Using  the  notation  of  359,  we  can  so  assign  A  that 

1       1 
a<A^    and    therefore    ->  — .       Let    a    be    assigned,    then 

11  "  ^    .  . 

2: <—«<-«,  and  ax<a^  or  ax  is  indefinitely  small. 
A        a 

Corollary  1.  The  quotient  of  an  indefinitely  small  number 
hy  any  number  not  indefinitely  small  is  indefinitely  small,  for 

if  a  =  — ,  the  preceding  result  is  —.  <a. 

a  a 

Corollary  2.  The  product  or  quotient  of  an  indefinitely 
great  number  by  any  number  not  indefinitely  small  or  great 


MISCELLANEOUS   TOPICS  307 

respectively  is   indefinitely  great,   for   from  a;  <  —  «  <  -  «,    it 

A        a 

1      A      a  1 

follows  that  ->  —  >-,  or  as  we  may  write  it,  since  -  =  JT, 
X      a       a  X 

-X'>-,  —  >-,  a'Jr>-,  where  a  is  not  indefinitely  great. 
a     a       a  a 

361.  Theorem.  Tlie  product  of  an  indefinitely  small  num- 
ber and  an  indejinitely  great  number  may  have  any  value. 

Proof.     If  a  is  not  indefinitely  great,  —  is  an  indefinitely 

^       ^       ax 

great  number  by  360,  and  x  x  ^  =  -,  which  may  have  any 

ax      a 

value  not  indefinitely  small.     Again  if  a  is  not  indefinitely 

small,  then  -—.  is  some   indefinitely  small  number  b}^  359 

11 

and  360,  Corollary  1,  and  X  x  ——  =  - ,  which  may  have  any 

aX      a 

value  not  indefinitely  great.     Hence  the  product  may  have 

any  value. 

Corollary.  The  quotieyit  of  tiuo  indefinitely  small  or  of  two 
indefinitely  great  numbers  may  have  any  value. 

362.  From  the  preceding  theorems  w^e  may  tabulate  the 
following  results,  x  and  y  being  two  indefinitely  small 
numbers,  and  X  and   Y  two  indefinitely  great  numbers. 

xy  is  indefinitely  small.  XY  is  indefinitely  great. 

a:—  is  indefinitely  small.  JT-  is  indefinitely  great. 


X  x 

xX\^  indeterminate.  yZ  is  indeterminate. 

—  is  indeterminate. 

y  Y 

X 

X  X 


—  is  indeterminate.  —  is  indeterminate. 

/  Y 

X   .    .  .  X 

-—  is  indefinitely  small.  —  is  indefinitely  great. 


308  COLLEGE   ALGEBRA 

363.  Theorem.  The  sum  of  n  indefinitely  small  yiumhers  of 
the  same  sign  is  indefinitely  small  provided  n  is  not  indefiriitely 
great. 

Proof.     Let  e^,  e^^  •••,  e^  be  ^  indefinitely  small  numbers; 

and  let  a  be  an  assigned  positive  number ;   -  is  also  an  as- 
signed  positive  number.     And 

n 

I     I  ^  ^ 

^2      <  -  ■> 

n 


\^n\<-' 

n 
Therefore  I  ^i  +  ^2  +  *"  +^n|<a»  (26) 

and  hence  ^1  +  ^2  +  " " '  +  ^« 

is  indefinitely  small.     But  if  n  be  indefinitely  great,  the  in- 

equality  i— ^  <  1,  which  before  held,  and  conditioned  the  con- 
a 

n 
elusion,  does  not  now  necessarily  hold,  for  by  361,  Corollary, 

^-^  may  have  any  value,  and  the  conclusion  no  longer  neces- 

n 

sarily  holds.     This  theorem  is  of  the  utmost  importance. 

Corollary.  The  sum  of  a  finite  number  of  indefinitely  small 
numbers  of  different  sig7is  is  indefiiiitely  small  or  else  zero. 

364.  Theorem.  The  sum  of  any  number  of  indefinitely 
great  numbers  of  the  same  sign  is  indefinitely  great;  but  if  the 
numbers  have  different  signs,  the  sum  is  iiideterminate. 


MISCELLANEOUS   TOPICS  309 

Proof.  In  the  first  case  X^,  X^,  •••,  X„,  being  indefinitely 
great, 

A 

and  since  I  Xj.  I  >  — , 

therefore,  |  Xj  |  +  |  Xg  |  +  •  •  •  +  |  X„  |  >  J.. 

For  the  second  case  X  may  be  any  positive  indefinitely 
great  nnmber,  and  a  any  positive  number  whatever.  Then 
X>A,  X+a:>A,  and  X-{-a=Y  is  indefinitely  great, 
Y—X=a,  X—  Y=  —  a,  and  the  difference  between  two 
infinitely  great  numbers  of  the  same  sign  may  have  any 
value  whatever.  Another  proof  of  this  might  be  given  as 
follows:  -,      -, 

y     X        xy 
which  by  363,  360,  and  361,  Cor.,  is  indeterminate. 

365.  Theorem.  The  difference  hetiveen  a  variable  x  and  its 
limit  a  is  an  indefinitely  small  number.  For  by  definition 
188  a  —  X  can  be  made  as  small  as  we  please,  that  is,  \a  —  x\<a^ 
where  l>a>0,  ov  x  —  a  —  e,  x=  a  +  e,  where  e  is  indefinitely 
small.     We  note  that  equivalent  statements  are  : 

x  =  a^  x—a  =  €^  x=a-{-€^  a-{-a>x>a  —  a,  a>x—a>  —  «,  Lx=a. 

366.  Theorem.  If  tivo  variables  are  constantly  equal  and 
each  api^roaches  a  limits  their  limits  are  equal. 

Proof.  Let  x  and  y  be  two  variables,  a  and  b  their 
limits.  Then  we  have  x=  a-\-  a,  y  =  b -\-  ^,  wdiere  a  and  fi 
are  indefinitely  small  and  x  =  y. 

Therefore  a  _  J  =  /3  -  «. 


310  COLLEGE   ALGEBRA 

Now  y8  —  «  is  either  indefinitely  small  or  else  zero  by  363, 
Cor.  It  cannot  be  indefinitely  small,  for  a  ~h  is  not  a 
variable,  therefore  it  must  be  zero;  therefore  a  =  h. 

367.  Theorem.  The  limit  of  the  algebraic  sum  of  a  finite 
number  of  variables  is  equal  to  the  algebraic  sum  of  their  limits. 

Proof.  Let  2^^  =  a^  +  e^,  x^  =  a^-\-  e^,  •  •  •,  x^  =  a^  +  e^, 
where  a?-^,  x^-,  •••,  x-n,  are  n  variables,  ^j,  a^,  •••,  a^,  their  limits, 
and  e^,  63,  •••,  e„,  n  indefinitely  small  numbers. 

Then 

x^-\rx^-}-  •  • .  -{-x^=  a^  +  a^-jr  •  •  •  +  a„  +€-^^-^e^-\-  •  •  •  +  e^. 

But  by  363,  Cor.,  €-^  +  e^-\-  •••  +  e,^  is  either  indefinitely 
small  or  else  zero  when  n  is  finite.  Therefore  by  defini- 
tion ^j  4- «2  +  •••  +^rt  is  the  limit  of  x-^-}-x^+  •••  +  x^^  and 

hence 

IjZx  =  2a  =  2.Lx. 

368.  Theorem.  The  liynit  of  the  product  of  a  finite  number 
of  variables  is  equal  to  the  product  of  their  limits. 

Proof.     Using  the  notation  of  367, 


=  .,a,.-a„(l+^)(l  +  ^ 


a„ 


M^-J  tto     *  *  *     ttj 


=  a-ittn  ••'  a 


\        ^"^  a-^     '^  a-^a^  "^  a-^  •  •  •  t«„ 


\       a-i  tt -1  •  •  •  tfjji 

(326) 


MISCELLANEOUS   TOPICS  311 

By  360  and  363  the  expression  in  the  parenthesis  is,  wlien 
n  is  finite,  indefinitely  small  unless  zero,  and  hence  by  360 
the  second  term  of  the  right-haud  member  is  indefinitely 
small  or  else  zero,  and  hence  by  definition  a-^a^  •••  <^n  is  the 
limit  of 


X 


^x^  ••'  Xfi^  or  Lx-^x^  •■■  x,i=  a^a^  •  •  •  (in=  Lx^Lx^  •  ■  •  Lx,^. 


369.  Theorem.  The  limit  of  the  quotient  of  tivo  variables  is 
equal  to  the  quotient  of  their  limits. 

Pkoof.     With  the  previous  notation 

x^      a^  -h  ej 

x^      a^      a^-\-  €^      a^       a^^i  —  a-^^ 
x^      a^~  a^-\-e.^      a^      ^2(^2  +  ^2) 

The  denominator  a^  (a^  +  ^2)  is  not  indefinitely  small  as 
long  as  a^^Q^  but  the  numerator  6^2^^  ~  <^i^2  ^^  indefinitely 
small  by  360  and  363,  Cor.,  unless  it  is  zero;  hence 

Z  -^  =  -i,  by  365,  or  X-i  =  y-^- 

Corollary.  If  two  variables  are  in  a  constant  ratio,  their 
limits  are  in  the  same  constant  ratio,  for 

XOC"{         Ct-t 

It  will  be  noticed  that  in  this  corollary,  as  well  as  in  other 
places,  it  is  convenient  to  speak  of  the  limit  of  a  constant  as 
that  constant. 


312  COLLEGE   ALGEBRA 

370.  EXAMPLES 

1.  Prove  that  the  limit  of  a  power  of  a  variable  is  equal 
to  that  power  of  the  limit  of  the  variable  when  the  exponent 
is  any  finite  real  constant. 

2.  The  limit  of  the  root  of  a  variable  is  the  root  of  the 
limit  of  the  variable. 

ON  THE   CONVERGENCE   AND   DIVERGENCE   OF  SOME 
PARTICULAR   SERIES 

371.  In  what  follows  generalizations  with  respect  to  the 
convergence  or  divergence  of  a  power  series 

I  ^Q  I  +  I  u^x  I  +  I  u^x^  I  +  •  •  •  +  I  u„x^^  I  +  •  •  • 

of  positive  terms,  where  ^f„  is  an  algebraic  function  of  n,  and 
related  problems  will  be  considered. 

372.  1.  Definition.  In  the  elementary  sense  an  algebraic 
function  of  a  variable  is  07ie  wJiieh  is  obtained  by  performing  a 
finite  number  of  operations  of  addition^  subtraction^  multiplica- 
tion^ division^  involutio7i,  and  evolution  on  the  variable.  In  the 
higher  sense  of  the  theory  of  functions^  x  is  an  algebraic  f mic- 
tion of  9^,  tvhen  x  ayid  n  are  connected  by  an  algebraic  equation 
F(x^  n)  =  0.  It  is  seen  that  the  second  sort  of  function 
includes  the  former  as  a  special  case. 

2.  Definition.  The  total  degree  of  an  elementary  algebraic 
functio7i  is,  as  has  already  been  stated  in  210,  the  degree  of 
the  numerator  miiius  the  degree  of  the  de7iominator,  e.g., 

(n  -\-  a')(n-\-  b)(n  -{-  <?) 


n{n  +  1)(^  +  2)(n  +  S)(n  +  4) 
is  of  total  degree  —  2. 


MISCELLANEOUS   TOPICS  313 

V?i  —  1 


1- 


is  of  total  degree  l  —  |  =  —  1. 

The  total  degree  of  a  higher  algebraic  function  61  the  sec- 
ond sort  is  defined  as  follows :  If  p  is  the  value  which  makes 

where  c  is  finite  and  not  equal  to  zero^  then  —  p  is  the  total 
degree  of  x=f(ji)^  obtained  from  the  relation  F(x^  n)  =  0. 
It  is  seen  that  this  definition  may  be  regarded  as  the  gener- 
alization of  the  former  and  includes  it.     (See  373,  2.) 

373.  LIMITS   OF   RATIOS 

1.  If  f(n)  is  an  algebraic  function  of  either  sort  of  total  de- 
gree zero^  its  limit  as  7i  =  oo  is  finite  and  not  zero.  When 
numerator  and  denominator  are  developed,  we  have  for  a 
function  of  the  first  sort,  t^      ^ 

^.   .  _    An'  4-  Bn'-'^  +  On'-*  +  .-•     _  n''      if 

-^^"^^  ~  A'n'  +  B'n'-''  +  C'n'-"  +  -'~  ^>  j^B[      C[  ' 


n"      n" 


whence,  L    fQii)  =—j  =  c, 

n  =  3o  J± 

where  c  is  finite  and  not  zero. 

If /(n)  is  a  function  of  the  second  sort,  this  proposition  is 
true  by  definition. 

2.  If  /(w)  is  an  algebraic  function  of  the  first  sort,  we 
have  the  following  theorem  :  That  2vhich  is  taken  as  defini- 
tion of  total  degree  for  a  function  of  the  second  sort  is  true  as 
a  property  of  a  function  of  the  first  sort.,  and  conversely. 

Proof,     hetf(n')  of  the  first  kind  be  of  total  degree  —  jt?, 


314  COLLEGE   ALGEBRA 

tlieii/(^)^^  is  of  degree  zero,  hence  its  limit,  as  ?^  =00,  is  by 
1  finite  and  not  zero.  Conversely,  \if(n)n^  has  a  limit  which 
is  finite  and  not  zero, /(92)  must  be  of  total  degree  —p^  as 
defined  for  a  function  of  the  first  kind,  for  the  assumption 
that  it  is  of  different  degree  leads  to  a  limit  0,  or  cx).  This 
identifies  the  definition  given  for  the  total  degree  of  an 
elementary  algebraic  function  in  372,  2,  as  a  special  case  of 
that  given  for  one  of  a  higher  kind. 

8.    If  f{n)  is  a  function  of  either  sort^  then 

L(mod)  /^"^^     =1. 

n  =  oo  f(n  —  1) 

For,  first,  since  the  total  degree  of   ^-^^  ^.     is  zero,  the  limit 

f(n  - 1) 

of  the  ratio  is  finite  and  not  zero.    Second,  if  —  jt>  be  the  total 

degree  of  f(n)  of  the  first  sort,  f(ii)iRP  is  of  degree  zero  and 

by  2  its  limit,  as  9^  =  00,  is  c.     Similarly,  since 

L  f(n-V)n^=    L  fQii-V)Qi-iy  =  ±c. 

«  =  00  W  —  1  =  ao 

Hence 
L  (mod)     -^^''^    =    L  (mod)     /(^^     =  (mod)  ±  -  =  1. 

If /(ti)  is  of  the  second  kind, 

L  (mod)/(9^)w^, 

•  n  =  <x> 

L  (mod)/(w  -  l}(n  -  1)^ 

n  — 1  =  00 

L  (mod-)  f(n  -  l)(n  -  1)^ 

«=oo 

and  L  (mod)  f(^ti  —  1) n^ 


MISCELLANEOUS   TOPICS  315 

are  identical,  the  first  three   obviously,  and  the  fourth  is 
equal  to  the  third  because 

4.    We  also  note  the  theorem  : 

L  •  J^  =  0,  or  oo, 

according  as p^^p^-,  tvhere  —pi  and  —p^  ^^^  ^he  total  degrees 
off  I  andf^. 

374.  THEOREMS   ON  CONVERGENCE 

1.  All  positive  series  ivhose  nth  terms,  u„^  t?„,  tv^,  •••,  a?'e  alge- 
braic functions  of  n  of  the  same  total  degree  are  convergent 
together  or  divergent  together.     For  let 


be  the  nth  terms  of  any  two  of  the  series.     Then  since 

=  c. 


L  /lOO 


where  c  is  finite  and  not  zero  (by  373,  1),  the  hypothesis  in 
207  is  satisfied,  and  therefore  the  conclusion  follows.  By 
means  of  this  theorem  we  may  classify  series  of  the  given 
kind  with  respect  to  the  total  degrees  of  their  nth  terms, 

and  since  the  series  whose  nth  term  is  v,^  =  ^  of  total  degree 

n^ 

—p,  is  convergent  if  j9  >1,  but  otherwise  divergent,  we  have 
the  following  theorem: 

2.  If  the  nth  term  of  a  positive  series  Un  =fO^^  *'^  ^^  alge- 
braic function  of  total  degree  —p-,  the  series  is  convergent  if 
jp  >  1,  otheriuise  it  is  divergent.  Thus  the  determination  of 
convergence  or  divergence  becomes  simply  a  matter  of  deter- 


316  COLLEGE   ALGEBRA 

mining  the  total  degree  of  u^^  and  this  can  often  be  done  by 
mere  inspection. 

3.  If  the  nth  term  of  a  positive  series  he  Uj^=f(ri)x^~'^^ 
where  f  is  an  algebraic  function  of  n^  then  the  series  is  conver- 
gent if  x<l^  divergent  if  x>l,  and  when  x=l^  convergent  if 
p>l,  otherwise  divergent^  where  —  p  is  the  total  degree  of 
f(n).  The  proof  of  this  theorem  follows  from  the  ratio 
test  (205,  206).     According  as 

I .  -^  >  1 


«iOO     lir 


< 


the  series  is  convergent  or  divergent.     When 


and 

L     ^"  =1, 

a  further  test 

is 

required. 

If 

J., 

or  if 

and 

L     '^'^  -1, 

n^ca  11, 


the  series  is  divergent.     We  have  here 

L  — —=  L     /^   ^     x  =  x 

n=(Xi  'If'n-l        n=ccj(^n        LJ 

(by  373,  3),  and  we  have  in  fact,  when  x<l  the  series  is  con- 
vergent, when  x>l  the  series  is  divergent,  and  when  x=l 
the  series  is  convergent  or  divergent,  according  as^^l. 


MISCELLANEOUS   TOPICS  317 

4.  In  case  of  a  series  of  positive  and  negative  terms  or  in 
case  of  a  series  with  complex  terms,  we  may  consider  the 
series  i      i  •  i        i  ■  i       9 1  ■         ■  1       « 1  ■ 

I  Wq  I  +  I  U^X  I  +  I  U2X'^  I  +  ■  •  •  +  I  '^n^    I  +  '"■> 

and  the  well  known  theorem  (214),  if  the  modular  series  of 
a  given  series  is  convergent,  the  series  is  also,  leads  to  other 
useful  applications.  From  what  has  already  been  said  it  is 
not  necessary  to  supply  further  details  for  the  treatment  of 
this  case. 

It  is  to  be  carefully  noted  that  in  the  foregoing,  factorial 
functions,  exponential  and  transcendental  functions,  and  all 
forms  of  higher  functions  have  not  been  considered. 

375.  EXAMPLES 

1.  The  series 

(l  +  a)(l  +  ^)(l  +  0  ,  (2  +  ^)(2+^)(2  +  0 
1.2.3.4.5  2.3.4.5-6 

■^  nQi  +  l)Oi  +  2){n  +  3)(n  +  4) 

is  seen  by  inspection  to  be  convergent  if  x<l,  divergent  if 
a;  >  1,  and  when  x  =  1^  convergent,  since  the  total  degree  of 
f(n}  is  -  2. 

2.  The  series 

12  71 


1+V2      I  +  2V3  1+nVn  +  l 

is  seen  at  once  to  be  divergent  since  the  total  degree  of /(w)  is 

1  _  3  =  _  1 

2  —         2* 


318  COLLEGE   ALGEBRA 

3.  The  series 

is  seen  in  the  same  way  to  be  convergent  if  a:  <  1,  diver- 
gent if  x>l^  and  when  x=l,  divergent  (total  degree  of 
f(n)  =  -l;  see  372). 

4.  The  series  whose  nth  term  is 


Un  =  n''(  Vw  —  1  —  2  Vw  —  2  +  Vn  —  o)x''\ 

is  convergent  when  x<l^  divergent  when  x>l^  and  when 
x=  1^  'd  little  calculation  shows  that  p  will  be  >  1,  if  /c  <  J  ; 
hence  Avhen  rr  =  1,  it  is  convergent  if  a:  <  ^,  otherwise  diver- 
gent. 

5.    The  series  whose  nth  term 

where /(ti)  is  obtained  from  the  cubic  equation 

a^a^  +  (^0^  +  h-^)x^  +  (Cq7i^  +  c-^7i  -\-C2)x  +  d^n^  +  d-^n'^  -f-  (^g^  +  f/g = 0, 

is  convergent  if  2  <  1,  the  modular  series  is  divergent  if  2;  >  1, 
and  when  2;  =  1,  the  modular  series  is  divergent,  since 

f(Yi)  =  X  and  =  GO  ,  as  7^  =  00  . 

To  show  this,  put  n  =  —  \ 

m 

the  equation  then  becomes: 
a^mV  +  (h^ifr?  +  h^)x^  +  {c^m  +  c^irp-  -f-  c^n^~)x  +  t7Q+  (i^m 

+  c?2^^^  +  d^  —  0, 
when  in  fact  if  m  —  0, 

all  three  roots  become  infinite. 


MISCELLANEOUS   TOPICS  319 

It  ^0  ^^     0  ^^  ^0  ^^      0' 

the  equation  approaches  the  form 

Qmx)^  +  (^mxy^  +  (^mx)  +  1  =  0. 
i.e.  mx  =  a^ 

where  a  is  a  complex  fourth  root  of  unity, 

x  =  an, 
hence /(w)  is  of  total  degree  1. 

6.    Given  F{x,  n)  =  7i^x^  +  4  nx"^  -f-  3  =  0, 

or  a;^  +  4  mV  -\-om^  =  0; 

as  m  =  0,  a;  =  0, 

Lff£Y  +  4f^):.  +  8)=0, 
m^\\mj  \mJ  J 


or 


L 

.             3^0) 
La:  = ^, 


where  0)^=  1, 

and  the  total  degree  of /(w)  is  -  1.     Hence  the  series  whose 
nth  term  is  \un\  =  \f{n) \  is  divergent. 

7.  F{x,  7^)  =  n^x^  +  2  nH^  +  2  =  0, 

or  a;3  +  2  ?7i%2  +  2  7?z6  =  0, 

as  m  =  0,  a:=  0. 

which  shows  that  x  is  of  total  degree  -  2.     Hence  the  series 
defined  by  u^  =f(ji)  =  x\^  convergent. 


320  COLLEGE   ALGEBRA 

8.  F(x,  n)  =  n^x^  +  3  nV  +  5  =  0. 

(^Y  +  3m-3(4y  +  5  =  0. 

x=f(n)  is  of  total  degree  —  |,  and  the  series  whose  nth. 
term  is  2/;^  =f(n)z"~'^  is  convergent  if  2;  ^  1. 

9.  Let  the  student  work  the  problems  of  215. 

376.    The  Product  of  Two  Infinite  Series. 
Theorem.     If  two  infinite  series 

JJ=  Uq-\-  u-^x  +  v^x^-]- h  Uj^x"^  +  •  •  • 

and  V=  Vq  +  v^x  +  v^x'^  +  •  •  •  +  v„a;"+  •  •  • 

are  absolutely  convergent^  then  the  third  infinite  series 

P  =  u^v^  +  (u^v^  +  u-^i^Q^x  +  (uqV^  +  11-^1^  +  ^2^0) -^"^  H 1"  (V« 

+  UiVn_i  H h  ^^,^'^0)2;"  +  •  •  • 

in  which  the  coefficient  of  any  po^ver  of  x  is  the  same  as  in  the 
product  of  U  and  V,  is  also  absolutely  convergent  and  equal 
to    UK 

We  only  need  to  prove  the  theorem  for  the  modular  series 
or  series  composed  of  the  absolute  values  of  the  terms  of  the 
given  series,  since  a  series  is  convergent  if  its  modular  series  is 
convergent,  by  214.  Denoting  the  modular  series  by  accents, 
we  have  ZZ'ga  X  V'^n  =  ^'2'i  +  terms  containing  2^"  and  higher 
powers  of  x  where  U'^w  V' ^n^  P\n  denote  the  sums  of  the  first 
2  n  terms  of  the  modular  series  of  u,  v,  p  respectively  ;  P' 2,n 
=  U'n  X  V'n  +  other  terms.  Hence  U' ^,,  V'^^  >  P'^„  >  U\  F'„; 
taking  the  limits  of  the  three  expressions  in  this  inequality, 
and  noting  that  L  U\j,  =  L  U\,  =  U\  L  W^j,  V\n  =  ^'  y\  etc., 
we  see  that  LP' ^^  =  LP' ,,  =  P'  =  U'V.  '^  Thus  L(iU\,,V' ^^ 
—  P^ji)  =  0,  and  since  U'^nV'^n-  P'2n^  U^^nV^n-  P^^n^  it  fol- 
lows that  X(  U^,,  V^„-  P2n)  =  0,  or  P  =  UV. 


MISCELLANEOUS   TOPICS  321 

377.    Vandermonde's  Theorem.     If  r,  s,  and  n  be  positive 
integers  such   that   r-{-s>n.  we  have    proved  in    162    that 


r+s 


^n  —  r^n~^r^n-l    '    s^l  +  r  ^n-2  *  •»  ^2  "^  "  *  "^  5^»* 


Multiplying  each  member  of  this  equation  hj  nl^  we  have 

''^A    T^  7-i     .    /^n\    7-»  7-»     ,  ,    /^ 


Denoting  „P^  by  n^,  this  result  takes  a  more  striking  form  : 
a  formula  which  could  be  obtained  from 


?% 


by  changing  all  the  exponents  into  subscripts.  Since  (2) 
is  of  degree  n  in  r  or  s,  and  since  it  is  true  for  more  than  n 
values  of  r  and  a  value  of  8  for  which  r  +  s  >  ?^,  it  is  true 
for  all  values  of  r,  and  in  the  same  way  it  can  be  shown  to 
be  true  for  all  values  of  s  and  a  value  of  r  by  83.  Hence 
it  is  true  for  all  values  of  r  and  for  all  values  of  s.  The 
identity 

(r  +  s)„  =  r,,  +  (jj  r„_iSi  +  Q)  ^»-2«2  +  "  *  +  ( J  ^» 
is  known  as  Vandermonde  s  TJieorem. 

378.    The  Binomial  Theorem  for  any  Index.     We  give  here 
another  proof  of  the  binomial  theorem,  see  221. 


Let 


/(^)^l  +  ^^+|^2^2+...  +  ^r^r^_....  (1) 


322  COLLEGE   ALGEBRA 

Then  /(^)=l  +  ^l^  +  ^^2_^...^^r^r_^...^  (2) 

and 

fQm+n)~l  +  ^  ^^x-{-^  ^2a:^H \-^ — y^a:^+  •••(3) 

J.  I  ^  I  T  I 


f(m)  and  /(9i)are  absolutely  convergent  series  when  |  a:  |  <  1 
(221),  and  the  coefficient  of  x^  in/(m)  xf(n}  is 

7-!      (/•- 1)!  1!      (r- 2)!  2!  (r-s)lsl  rV 

If       .         .  r! 


r  I  V  (r  —  s) !  s  I  / 

which  by  Vandermonde's  Theorem  is  equal  to  ^^ — — — ^.  (4) 

Thus  the  coefficients  of  the  different  powers  of  x  in  f(jri) 
xf(n)  are  always  equal  to  the  coefficients  of  the  correspond- 
ing powers  of  x  in  f(rn  +  ti),  and  therefore  by  376 

/(m)  xf(n)  =f(m  -}-  n),  (5) 

provided  |  a^  |  <  1. 

By  using  (5)  repeatedly  we  obtain 

/(m)  xf(n)  xfQp)  x  •.•=/(m  +  n  +  ^  +  •••)•         (^) 

Let  m  —  n—'p—  ••.=-,  r  and  s  being  positive  integers,  and 

s 

there  being  s  of  the  numbers  w,  ?2,  jo,  •••,  and  (6)  becomes 


[• 


^e 


=/?:x.   =/(r).  (7) 


But  when  r  is  a  positive  integer,  /(r)  =  (1  +  xY' 


MISCELLANEOUS  TOPICS 


323 


Therefore 


and 


•'li 


=  0-  +  xy, 


(i  +  .)^=/g 


(8) 


(9) 


This  proves  the  binomial  theorem  for  a  positive  fractional 
commensurable  exponent.  By  limits  as  in  221  the  proof 
can  be  extended  to  any  positive  exponent. 

Again  /(0)  =  1,  and  by  (6)/(-n)  x/(n)=/(0)  =  l.  (10) 


Therefore  /(— 7i)  = 


Y 

A 


f.  .  =  7T^^  =  a  +  ^)-"by(9)and91. 
f(n)     (1  +  xy 

Hence  (1  +  a;) "»=/(- w),  (11) 

which  proves  the  theorem  for  any  negative  exponent.    Hence 
the  theorem  is  true  for  any  real  exponent  provided  |  a:|<l. 

379.  The  Complex  Variable  as  Function  of  its  Modulus  and 
Argument.  We  have  seen  in  126, 129  that  the  complex  variable 
z  =  X  -\-  yi  may  be 
represented  by  the 
point  P  =  (x,  y)  in 
the  xy  plane  ;  that 
to  every  value  of  z 
there  corresponds  a 
single    point    P  = 

(a;,  ^),     and     con-     

versely     to      every 

point     P  =  (a:,    y') 

there  corresponds  a 

single    value    of   2, 

z  =  x-\-  yi^  and  that 

z  may  be  expressed  as  2;  =  r(cos  <\>  -{- i  sin  c^),  where,  Fig.  34, 

r  =  V2;2  +  ^/2  is   the    modulus   of   z,    and  ^  is    its    argument 

or  amplitude. 


o 


X 


Fig.  34. 


324  COLLEGE   ALGEBRA 

When  the  modulus  of  a  complex  variable  vanishes^  the  variable 
vanishes  and  conversely.  Yov  if  2;  =  0,  r(cos  <^-\-  i  sin  <^)  =  0, 
and  either  r  =  0,  or  cos  </>  +  ^  sin  (/>  =  0  ;  but  it  is  impossible 
for  cos  (^  +  ^  sin  <^  to  vanish,  for  cos  (/>  and  sin  <^  do  not  vanish 
together  ;  therefore  the  vanishing  of  z  is  due  to  the  van- 
ishing of  r  alone.  This  is  seen  geometrically  in  that  z=0 
represents  the  point  0,  or  the  origin,  and  only  for  the  point 
0,  or  the  origin,  does  z  vanish. 

Similarly  a  complex  variable  z  is  infinite  when  and  only 
when  its  modulus  is  infinite. 

380.    De  Moivre's  Theorem  for  a   Positive  Integral  Index. 

We  shall  prove  here  that  for  a  positive  integer  m,  (cos  cc  -{- 
i  sin  a)^  =  cos  ma  +  i  sin  ma.  This  can  also  be  proved  for  any 
real  index.  The  theorem  is  known  as  De  Moivre's  Theorem. 
The  proof  for  a  positive  integer  follows  at  once  from  131, 
where  it  is  shown  that  in  the  product  of  two  complex  num- 
bers, the  modulus  of  the  product  is  equal  to  the  product  of 
the  moduli  of  the  factors,  and  the  argument  of  the  product 
is  equal  to  the  sum  of  the  arguments  of  the  factors,  two  fac- 
tors being  given.  If  three  factors  are  given,  the  product  of 
two  of  them,  obtained  by  this  principle,  may  be  regarded  as 
a  single  factor,  and  hence  the  principle  for  two  factors  ap- 
plies with  the  same  conclusions  as  before  extended  to  three 
factors.  And  in  a  similar  way  we  obtain  for  any  number  of 
factors  : 

The  modulus  of  the  product  is  equal  to  the  product  of  the 
moduli  of  the  factors. 

The  argument  of  the  product  is  equal  to  the  sum  of  the 
arguments  of  the  factors. 

Applying  these  principles  to  the  present  case,  we  have  the 
result,  for  the  number  of  factors  is  m^  the  modulus  of  each 
factor  is  unity,  and  the  argument  of  each  factor  a  ;  hence 


MISCELLANEOUS   TOPICS  325 

the  modulus  of  the  product  is  unity  and  its  argument  a+  a 
+  •  •  •  +  «  to  m  terms  =  ma.    Therefore  for  a  positive  integer  jn 

(cos  a  +  i  sin  «)"^  =  cos  ma  +  i  sin  ma. 

381.  Continuity  of  a  Function  of  a  Complex  Variable.  As 
in  the  case  of  a  function  of  a  real  variable  319,  so  we 
define  continuity  for  a  function  of  a  complex  variable,  and 
say  in  particular :  A  rational  integral  algebraic  function 
/(s;),  where  z=  x  -{-  yi,  is  continuous  at  the  finite  value 
z,  or  at  the  point  (x^  ?/),  when  it  satisfies  the  condition 
|/(2J  +  A)  — /(2)  I  =  0,  as  I A  I  =  0,  in  whatever  direction  around 
the  point  (a;,  ?/),  h  may  be  taken. 

It  is  thus  seen  that/(2;)  =  a^z^  +  a^z'"''^  +  •••  +  ««  is  contin- 
uous, for  /(2)  is  single-valued,  and  the  development  of  317 
holds  for/(^  +  A),  and 

/(^  +  /0-/(z)=A/'(^)  +  ^/"(2)+  ...  +->"(2). 

Z.  71. 

As  I  A  1=0,  |/»|.[/'(2)|+l|il  -I/"  0)1+  -  =0 

by  315.     A  fortiori 

I  ¥'(^}  +  |t/"C^)  +  •  •  •  I  =  !/(^  +  '0  -/(^)1  =  0, 

since  the  modulus  of  a  sum  of  terms  is  smaller  than  or  at 
most  equal  to  the  sum  of  the  moduli  of  the  terms. 

382.  Geometrical  Representation  of  a  Function  of  a  Com- 
plex Variable.  When  f{z)  =  a^z" -]-a^z'*~'^ -^  •••  +  a„  is  ex- 
panded, it  is  seen  that  w  =fQz')  =  w  +  vi^  where  u  =f^(x.,  ?/), 

Just  as  z  represents  a  point  (x,  ?/),  so  ^v  represents  a  point 
(t*,  v^.  For  distinctness  of  representation,  we  shall  use  two 
planes,  Figs.  35  and  36,  one  the  z  plane  or  xy  plane  for  repre- 


326 


COLLEGE   ALGEBRA 


senting  the  values  of  2,  and  the  other,  the   w  plane  or  uv 
plane  for  those  of  w. 

From  the  continuity  of  w  just  proved,  381,  we  see  that  if 
the  point  P  representing  z  traces  a  curve  in  the  x^  plane,  the 


Y 
A 


0 


X 


Fig.  35. 


point  Q  representing  w  traces  another  curve  in  the  uv  plane. 
That  w  =f(z)  vanishes  is  the  same  as  to  say  that  when  z  is 
at  P  =  (x,  y\  Qis  at  0' =  (0,  0). 


o' 


u 


Fig.  36. 


383.   Isogonality  of  the  Function  f{z). 

When  z  starts  from  P  and  traces  the  curves  Pa,  P5, 
Fig.  37,  w  starts  from  Q  and  traces  the  corresponding  curves 
Qa' ^  Qh\  Fig.  38,  and  we  shall  prove  that  if  f'(z)  is  neither 


MISCELLANEOUS  TOPICS 


327 


Fig.  37 


0  nor  00  at  ft  the  angle  t\Qt'^  between  the  tangents  to  the 
curves  Qa',  Qb',  at  ft  is  equal  to  the  angle  t^Pt^  between  the 


Fig.  38. 


328  COLLEGE  ALGEBRA 

corresponding   tangents   at    P,    and   if  /'(^)=/'^(2;)=  •••  = 
/(->(2;)  =  0, /(-+^)(^)^  0,  ^  t\Qt'^  =  (m  +  1)  ^  t^Pt^, 

By  128,  130, 

OP      ,       Oa        ,  Oa-OP      Pa       ,      .         ,   , 

Oa! 
similarly   w'  =  w  -\-k,  where  k  =  ^^^,, 

and  Oir=Oif'  =  +  l. 

Also      1^1  =  ]^''^f'  f;^=  length  Pa  ;  |A:|  =  length  G«^ 
length  C>jM 

and  as         a  =  P,  |7i|  =  0,  a' =  ^,  and  |^|  =  0. 

w'=f(z')=fCz  +  JO, 

or  if  p  and  r  are  the  moduli  and  a  and  (j>  the  arguments  of  k 
and  A  respectively, 

r(cos9+z  sm  9)  Zl 

Taking  the  limits  of  both  members  as  a  =  P,  i.e.  as  |^|  =0, 
and  observing  that  (j)  =  yjr^  «  =  A  we  have 

\     r  J  \GOS  yjr  +  z  sin  sjrj 
or         rX^ycos(/3-i/r)+zsin(^-'>/r))=/'(2)  by  132. 


MISCELLANEOUS   TOPICS  329 

Ph      Oh' 
Similarly,  if  h'  and  k'  are  -— -,  -^,  respectively,  i/r'  and  ^' 

the  corresponding  angles  to  the  tangents  Pt^,  Qt\,  r'  and  p^ 
the  moduli  of  h'  and  k', 

(i2^^(cos  (^'  -  ^')  +  isin  (/3'  -  ^/r'))  =  /'(^). 

Hence  mod /'(«)  =  i2,  or  i^,  i?- =  ie^, 

and  if/'(0)  is  neither  0  nor  oo,  the  arguments  /3  —  yjr  and 
/3'  —  yjr'  must  be  equal,  i.e. 

^-ylr=/3'  -f.or  13'- I3=ylr'-ylr,  i.e. 
^t',Qt'^  =  ^t,Pt,^. 

This  is  called  the  isogonal  property,  or  isogonality^  of  the 
function /(s;). 

384.    Failure  of  the  Isogonal  Property. 

the  isogonal  property  fails,  but  is  replaced  by  another  angle 
property. 

In  this  case  k  =  —^ /(»^+i>  (2;)  +  •  •  •, 

^  1        /("^+i)(2)  +  ..., 


/(m+l)(^). 


Similarly      L-f—= ^ f"'^^Kz). 

^  h'"'+'     (>?i  +  l)I  ^  ^ 


.  /^   jQ   A     (cos /3 +  2  sin /3)    ^fj^    p'   \     (coi^^'-hism/3') 


330  COLLEGE   ALGEBRA 

or,  by  De  Moivre's  Theorem,  and  132, 

(^  ;£l)(^^^  (^  -  (m  +  1)  t)  +  i  sin  (^  -  (w  +  1)  ./r)^ 

=  (-^  /Cri)(^^<^'  -  (m  +  1)  ^/^O  +  ^  sin(/3^  -  (m  +  1)  ^/r')^ 
=/('"+!>  (2). 

.*.  L-^—;  =  L   ;      .  and  since  f^^+^Y^^  is  neither  0  nor  00,  the 

arguments  /3  —  (m  +  1)  i/r  and  y8'  —  (m  +  1)  t/t'  are  equal, 

whence  /3'  -  y8  =  (m  +  1)  (i/r'  -  i/r), 

or,  ^  f  \  ^^'2  =  (^  +  1)  ^  hPt^. 

385.  The  Proof  that  Every  Algebraic  Polynomial  a(^"-{-a^z"~'^ 

-\ f-fl/j  has  a  Root.     We  prove  this  proposition,  known  as 

tlie  fundamental  proposition  of  algebra,  by  indirect  reasoning. 
If  possible,  let  tliere  be  no  value  of  z  for  which  f(z^  =  0. 
Then  it  follows  that  there  must  be  a  value  of  /(^),  repre- 
sented by  Q  and  corresponding  to  a  value  of  z^  represented 
by  P,  and  having  a  modulus,  such  tliat  for  no  other  value  of 
f(z)  is  the  modulus  smaller ;  that  is  for  no  other  value  of 
f(z)  can  the  point  representing /(2)  be  nearer  the  origin  0' , 
We  consider  two  cases  : 

(1)  /(^)  ^  0, 

(2)  f\z~)  =f'(z')  =  ...  =f^^-\z')  =  0,  /(-+i)(^)  ^  0. 

1.  Since  for  the  point  Q,  f'(z)  is  neither  0  nor  00,  and 
since  f(z)  represented  by  Q  is  continuous  at  §,  if  z  moves 
5  out  in  all  posible  directions  on  rays  from  P  once  around, 
Fig.  39,  IV  by  virtue  of  its  continuity  and  isogonality,  381, 
383,  will  move  out  in  all  possible  directions  from  Q  once 
around.  Fig.  40,  and  therefore  will  somewhere  move  nearer 


MISCELLANEOUS  TOPICS 


331 


to  the  origin  than  Q  itself.  Hence  the  conclusion  that  there 
is  a  value  of  f(z)  having  a  modulus  such  that  for  no  other 
value  of /(2!)  is  the  modulus  smaller,  is  false,  and  hence  also 


I 


0 


X 


Fig.  39. 


the  premise  that  there  is  no  value  of  z  for  which  f(z)  =  0, 
from  which  this  conclusion  followed.  Therefore  for  some 
value  of  2,   «q2^  + a^a;""^  +  •••  +  a;j  vanishes. 


V 


O' 


u- 


Fig.  40. 


2.    In  this  case  we  cause  z  to  move  out  from  P,  until  the 

rays  from  P  cover  a  sector of  the  way  around,  then 

m  +  1 

by  384,  to  will  have  moved  out  from  Q  in  all  possible  direc- 
tions once  around,  and  the  result  is  the  same  as  in  case  1. 
Hence  a^z^^  +  a-^z^~'^-\-  •••  -\-  a^  always  has  a  root. 


INDEX 

(The  numbers  refer  to  pages) 

Abscissa,  2. 

Algebraic  polynomial,  root  of,  330. 

Amplitude  of  complex  number,  85,  88. 

Antecedent,  20. 

Argument  of  complex  number,  88. 

Arithmetical  progression,  95. 

Auxiliary  series,  157. 

Axes,  2. 

Base  of  a  system  of  logarithms,  217. 
Binomial  coefficients,  126,  130. 

series,  convergency  of,  171. 
surd,  extraction  of  square  root  of,  73. 
theorem,  any  real  exponent,  169,  321. 
general  term,  129. 
greatest  term,  133. 
positive  integral  exponent,  124. 
Binomial  quadratic  surds  enter  equations  in  pairs,  268. 
Biquadratic  equation,  290. 

Characteristic  of  a  logarithm,  222. 
Circle,  coordinates  of  center,  8. 
equation  of,  6,  7. 
radius  of,  8. 
Coefficients,  binomial,  126,  130. 
Cologarithms,  227. 
Combinations,  113. 

complementary,  116. 
Commensurable  numbers,  50. 
Complex  number,  52,  79. 

amplitude  of  a,  85,  88. 

argument  of  a,  88. 

conjugate  of  a.  80. 

graphical  representation  of  a,  82,  85. 

modulus  of,  80. 
Complex  numbers,  identity  theorems  for,  81. 

333 


334  INDEX 

Complex  roots  enter  equations  in  pairs,  268. 

Complex  variable  as  function  of  its  modulus  and  argument,  88,  323. 

continuity  of,  325. 
Compound  ratio,  20. 
Conjugate  complex  numbers,  80. 
Conjugate  trinomial  surds,  71. 
Consequent,  20. 
Constants,  139. 
Continued  fractions,  185. 

convergents  of,  187. 

alternately  greater  and  less,  194. 
closer  and  looser  limits  of  error  of,  195. 
lowest  terms,  in  their,  194. 
recurring,  188. 

root  of  quadratic  equation,  188. 
Continuity  of  a  function  of  a  complex  variable,  325. 
Convergency  and  divergency  of  series,  147. 
Convergency  of  binomial  series,  171. 

geometrical  series,  150. 
some  particular  series,  312. 
Coordinates,  2. 
Cube  roots  of  unity,  271. 
Cubic  equation,  solution  of,  287. 
Cubic,  reducing,  291. 
Curve,  to  plot,  4. 

Decomposition  of  fractions,  173. 
De  Moivre's  theorem,  324. 
Derived  functions,  264. 

geometrical  interpretation  of,  280. 
Descartes'  rule  of  signs,  267. 
Determinants,  244*. 

column  of,  245. 
elements  of,  246. 

conjugate,  246. 
self-conjugate,  246. 
minors  of ,  253. 

first,  254. 
principal  diagonal  of,  246. 
product  of  two,  259. 
row  of,  245. 

secondary  diagonal  of,  246. 
solution  of  linear  equations  by,  256. 
terms  of,  246. 

principal  or  leading,  246. 


INDEX  335 

Development  of  a  fraction  into  a  series,  1G6. 

general  term  in,  182. 
Development  of  a  function,  263. 
Differences,  finite,  206. 

orders  of,  207. 
Discriminant  of  cubic,  289. 

of  quadratic,  38. 
Distance  between  two  points,  2,  8. 
Duplicate  ratio,  21. 

Equal  roots  of  equations,  36,  289,  294. 
Equation  of  locus,  2. 
Equations  of  first  degree,  6. 

graph  of,  4. 

represent  straight  lines,  6. 
higher  degree,  6. 

graphs  of,  6. 
quadratic,  theory  of,  32. 
radical,  74. 

extraneous  roots  of,  77. 
solutions  by  determinants,  256. 
theory  of,  262. 
Error,  closer  and  looser  limits  of,  195. 
Euler's  cubic,  291. 
Exponential  function,  231. 
theorem,  241. 
Exponentiation,  217. 
Exponents,  theory  of,  52. 
Extraneous  roots  of  radical  equations,  77. 

Factorial  7i,  111. 

Factoring  of  quadratic  expressions,  34. 

symmetric  and  related  expressions,  272. 
Factor  theorem,  44. 
Finite  differences,  206. 

orders  of,  207. 
Fractions,  continued,  185. 

convergents  of,  187. 
infinite,  185. 
recurring,  188. 

root  of  quadratic  equation,  188. 
terminating,  185. 
decomposition  of,  173. 
development  of,  into  series,  166. 
partial,  173. 


336  INDEX 

Function,  derived,  264. 

development  of,  263. 

exponential,  231. 

generating,  205. 

graphic  representation  of,  1. 

isogonality  of,  326. 

failure  of,  329. 

of  a  complex  variable,  geometrical  representation  of,  325. 

symmetric,  271. 
Fundamental  proposition  of  algebra,  330,  331. 

General  term  in  the  development  of  a  fraction,  182. 
Generating  function  of  a  recurring  series,  205. 
•Geometric  addition,  86,  89. 
division,  91. 
multiplication,  86,  90. 
subtraction,  89. 
Geometric  interpretation  of  the  derived  function,  280. 
Geometrical  progression,  99. 

convergency  of,  150. 
Graph,  4. 

of  equation  of  first  degree,  4. 

higher  degree,  6. 
Graphic  representation  of  a  function,  1. 

a  point,  1. 

complex  numbers,  82,  85. 
direct  variation,  28. 
inverse  variation,  29. 
real  and  complex  roots,  39,  40. 
Graphical  solution  of  simultaneous  equations,  11. 

Harmonical  progression,  103. 
Harmonic  division,  104. 

series,  158. 
Horner's  method,  297. 

Identity  of  two  polynomials,  46. 
theorem,  46. 

theorems  for  complex  numbers,  81. 
Imaginary  numbers,  51. 
Indefinitely  great  numbers,  305. 
small  numbers,  305. 
Indeterminate  equations  of  the  first  degree,  196. 

general  solutions  of,  197. 
particular  solutions  of,  197. 


INDEX  337 


Indeterminate  forms,  142,  807. 
Indices,  tlieory  of,  52, 
Induction,  matliematical,  125,  302. 
Inequality,  13. 

definitions,  13. 

fundamental,  17. 

in  same  or  opposite  sense,  14. 

members  of,  13. 

notation,  13. 
Infinites,  305. 
Infinitesimals,  305. 
Interpolation  formula,  213. 
Inverse  ratio,  21. 

Involution  and  evolution  of  surds,  67. 
Irrational  numbers,  58. 
Isogonality,  325. 

Limits,  139,  305. 
Linear  equation,  0. 
Locus,  4. 

equation  of,  3. 
symmetrical,  6. 
Logarithmic  series,  241. 
Logarithms,  217. 

Briggs,  221. 
calculation  of,  242. 
characteristic  of,  222. 
common,  221. 
mantissa  of,  222. 
Napierian,  221. 
natural  system  of,  221. 

base  of,  237. 
numbers  corresponding  to,  found  from  table,  226. 
of  numbers  found  from  table,  223. 
table  of,  224,  225. 

use  of,  223. 

tabular  difference  of,  226. 

Mantissa  of  a  logarithm,  222. 
Mathematical  induction,  125,  302. 
Minors  of  determinants,  253. 
Modular  series,  161. 
Modulus  of  a  complex  number,  80. 
Multinomial  theorem,  134. 

fireneral  term  of,  134. 


338  INDEX 

Napierian  base,  237. 
Natural  system  of  logarithms,  221. 
Number  of  real  roots  of  an  equation,  292. 
Numbers,  commensurable,  50. 

complex,  52,  79. 

incommensurable,  51. 

rational  and  irrational,  58. 

real  and  pure  imaginary,  51. 
Numerical  equations,  296. 


Ordinate,  2. 

Origin,  2. 

Oscillating  series,  150,  151. 


Partial  fractions,  173. 
Path,  equation  of,  3. 
Permutations,  108,  109. 
Plotting  a  point  or  curve,  4. 

a  straight  line,  6. 
Point,  coordinates  of,  2. 

representation  of,  1. 
Points  of  intersection  of  two  curves,  7. 
Polynomials,  identity  of  two,  46. 
Product  of  two  infinite  series,  320. 
Progression,  arithmetical,  95. 

common  difference  of,  95. 
means,  97. 
geometrical,  99. 

constant  ratio  of,  99. 
means,  101. 
harmonical,  103. 

means,  103. 
Proportion,  22. 

by  addition,  23. 
by  alternation,  22. 
by  composition  and  division,  24. 
by  inversion,  23. 
by  subtraction,  24. 
continued,  24, 
extremes  and  means  of,  22. 
Proportional,  mean,  25. 
third,  25. 


INDEX  339 

Quadrants,  2. 

Quadratic  equations,  discriminant  of,  38. 

formation  of,  with  given  roots,  33. 
roots  of,  32. 

graphical  representation  of  real  and  complex, 

39,  40. 
nature  of,  36. 
two  and  only  two,  41. 
zero  and  infinite,  36,  37. 
solution  of,  by  inspection,  35. 
theory  of,  32. 
Quadratic  expressions,  factoring  of,  34. 
surds,  properties  of,  72. 

Radical,  entire,  58. 

equations,  74. 
mixed,  58. 
Radicals,  57. 
Ratio,  20. 

antecedent  of,  20. 
compound,  20. 
consequent  of,  20. 
duplicate,  21. 
inverse,  21. 
of  greater  inequality,  20. 

lesser  inequality,  20. 
subduplicate,  21. 
sub  triplicate,  21. 
terms  of,  20. 
triplicate,  21. 
unit,  20. 
Rational  number,  68. 
Real  numbers,  51. 
Recurring  series,  203. 

generating  function  of,  205. 
order  of,  203. 
scale  of  relation  of,  203. 
sum  of  n  terms  of,  204. 
Reducing  cubic  of  biquadratic,  291. 

quadratic  of  cubic,  287. 
Remainder  after  n  terms  in  a  series,  148. 

theorem,  48. 
Representation  of  a  point,  1. 

function,  1,  325. 
Rolle's  theorem,  282. 


340  INDEX 

Root,  square,  50. 

every  equation  of  the  nth  degree  has  a,  330. 
Roots,  every  equation  of  tlie  ?ith  degree  has  w,  45. 
extraneous,  77. 
of  a  quadratic  equation,  32,  36. 

two  and  only  tw^o  of,  41. 
zero  and  infinite  of,  36,  37. 

Scale  of  relation,  203. 
Series,  147. 

absolutely  convergent,  147,  161. 

auxiliary,  157. 

convergency  of,  148-163,  171,  312,  320. 

convergent,  147. 

divergent,  147. 

exponential,  241. 

finite,  147. 

harmonic,  158. 

infinite,  147.   - 

the  product  of  two,  320. 

logarithmic,  241. 

modular,  161,  317. 

necessary  and  sufficient  conditions  for  convergency  of,  148,  149. 

oscillating,  150,  151. 

recurring,  203. 

scale  of  relation  of,  203. 

remainder  of,  after  n  terms,  148. 

summation  of,  183,  199,  303. 

tests  for  convergency  of,  151. 
Simultaneous  equations,  7. 

geometrical  representation  of,  7. 
solution  of,  by  graphical  methods,  10,  11,  12. 
Solution  of  biquadratic  equation,  290. 
cubic  equation,  287. 
quadratic  equations  by  inspection,  35. 
Sturm's  theorem,  292. 

functions,  292,  298. 
Summation  of  series,  183,  199,  303. 
Surds,  57. 

addition  of,  62. 

comparison  of,  67. 

division  of,  66. 

entire,  58. 

involution  and  evolution  of,  67. 

mixed,  58. 


INDEX  341 


multiplication  of,  63. 

quadratic,  properties  of,  72. 

square  root  of  binomial,  73. 

rationalization  of,  66-68. 

similar,  61. 

subtraction  of,  62. 
Symmetrical  locus,  6,  7. 
Symmetric  functions,  271. 

Table  of  logarithms,  224,  225. 
Tabular  difference,  226. 
Theory  of  equations,  262. 

indices,  52. 

quadratics,  32. 
Total  degree,  158,  313. 

Undetermined  coefficients,  164. 

Vandermonde's  theorem,  321. 
Variables,  139. 
Variation,  28. 

direct,  28. 

inverse,  28,  29. 

joint,  28,  29. 


{/ 


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